# Problem of the Month (August 2000)

A Friedman number is a positive integer which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. For example, 25 = 52 and 126 = 21 × 6. The Friedman numbers are sequence A036057 of the Encyclopedia of Integer Sequences.

All Friedman numbers with 4 or fewer digits are known:

` 25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024,1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349,2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592 ,2737, 2916, 3125,3159, 3281, 3375, 3378, 3685, 3784, 3864, 3972, 4088, 4096, 4106, 4167, 4536, 4624,4628, 5120, 5776, 5832, 6144, 6145, 6455, 6880, 7928, 8092, 8192, 9025, 9216, 9261.`
Determine why each of these is a Friedman number. You might also have fun confirming that 123456789 and 987654321 are Friedman numbers. Find some more Friedman numbers. What else can you show about Friedman numbers? If F(n) is the number of Friedman numbers less than n, can you show F(n)/n → 1? or even disprove F(n)/n → 0?

Mike Reid thought that Friedman numbers are "nicer" if the digits are used in the proper order. These are shown in red below.

Mike Reid, Ulrich Schimke, and Philippe Fondanaiche solved all the Friedman numbers with 4 or fewer digits. Here are the solutions:

Friedman Numbers With 4 or Fewer Digits
 25 = 52 121 = 112 125 = 51+2 126 = 6 × 21 127 = – 1 + 27 128 = 28–1 153 = 3 × 51 216 = 62+1 289 = (8 + 9)2 343 = (3 + 4)3 347 = 73 + 4 625 = 56–2 688 = 8 × 86 736 = 7 + 36 1022 = 210 – 2 1024 = (4 – 2)10 1206 = 6 × 201 1255 = 5 × 251 1260 = 6 × 210 1285 = (1 + 28) × 5 1296 = 6(9–1)/2 1395 = 15 × 93 1435 = 35 × 41 1503 = 3 × 501 1530 = 3 × 510 1792 = 7 × 29–1 1827 = 21 × 87 2048 = 84 / 2 + 0 2187 = (2 + 18)7 2349 = 29 × 34 2500 = 502 + 0 2501 = 502 + 1 2502 = 2 + 502 2503 = 502 + 3 2504 = 502 + 4 2505 = 502 + 5 2506 = 502 + 6 2507 = 502 + 7 2508 = 502 + 8 2509 = 502 + 9 2592 = 25 × 92 2737 = (2 × 7)3 – 7 2916 = (1 × 6 × 9)2 3125 = (3 + 1 × 2)5 3159 = 9 × 351 3281 = (38 + 1) / 2 3375 = (3 + 5 + 7)3 3378 = (7 + 8)3 + 3 3685 = (36 + 8) × 5 3784 = 8 × 473 3864 = 3 × (– 8 + 64) 3972 = 3 + (9 × 7)2 4088 = 84 – 8 – 0 4096 = (4 + 0 × 9)6 4106 = 46 + 10 4167 = 46 + 71 4536 = 56 × 34 4624 = (64 + 4)2 4628 = 682 + 4 5120 = 5 × 210 5776 = 767–5 5832 = (2 × 5 + 8)3 6144 = 6 × 44+1 6145 = 6 × 45 + 1 6455 = (64 – 5) × 5 6880 = 8 × 860 7928 = 892 – 7 8092 = 902 – 8 8192 = 8 × 29+1 9025 = 952 + 0 9216 = 1 × 962 9261 = 219–6

In 2002, I wrote a program capable of finding all the 5–digit Friedman numbers. Here they are. In 2013, Giovanni Resta found an omission.

5–Digit Friedman Numbers
 10192 = 1012 – 9 10201 = 1012 + 0 10251 = 51 × 201 10255 = 5 × 2051 10368 = 8 × 61+0+3 10426 = 26 × 401 10521 = 21 × 501 10525 = 5 × 2105 10575 = 15 × 705 10824 = 1042 + 8 (TG) 10935 = 15 × 93 + 0 11025 = (110 – 5)2 11163 = 3 × 611+1 11259 = 9 × 1251 11264 = 11 × 26+4 (MR) 11439 = 9 × 31 × 41 11663 = 16 × 36 – 1 11664 = 1 × 1 × 66 / 4 11665 = 66 / (5 – 1) + 1 11844 = 84 × 141 11848 = 8 × 1481 11943 = 9 × (113 – 4) 12006 = 6 × 2001 12060 = 6 × 2010 12091 = 1102 – 9 (PF) 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12104 = 1102 + 4 12105 = 1102 + 5 12106 = 1102 + 6 12107 = 1102 + 7 12108 = 1102 + 8 12109 = 1102 + 9 12167 = (16 + 7)1+2 (PF) 12288 = (1+2) × 88/2 12321 = (113 – 2)2 (TG) 12337 = 73 × 132 12384 = 3 × 4128 12493 = (4 + 9) × 312 12505 = 5 × 2501 12544 = (51 – 2) × 44 12546 = 51 × 246 12550 = 5 × 2510 12595 = 5 × 2519 12600 = 6 × 2100 12762 = 6 × 2127 12768 = 8 × 21 × 76 12769 = (96 + 17)2 12798 = 2 × 79 × 81 12802 = 2 × (802 + 1) 12843 = 3 × 4281 12850 = (1 + 28) × 50 12955 = 5 × 2591 12960 = 160 × 92 12964 = 14 × 926 12996 = (6 × (1 + 9 + 9))2 13125 = 21 × 53+1 13176 = 61 × (7 – 1)3 13225 = (1 × 5 × 23)2 13286 = 26 × (83 – 1) 13243 = 41 × 323 13496 = 4 × ((6 + 9)3 – 1) 13545 = 3 × 4515 13689 = (9 × 13)8–6 13725 = 5 × ((2 × 7)3 + 1) 13764 = 6 × 31 × 74 13813 = (3 × 8)3 – 11 13822 = (3 × 8)2+1 – 2 13823 = (3 × 8)3 – 2 + 1 13824 = (3 × 8)4+1–2 13825 = 1 + (3 × 8)–2+5 13826 = (18 + 6)3 + 2 13832 = (3 + 21)3 + 8 13842 = 243 + 18 13950 = 15 × 930 14035 = 35 × 401 14129 = 114 – 29 14168 = 11 × (64 – 8) 14175 = 7 × 451+1 14256 = (2 × 5 + 1) × 64 14350 = 35 × 410 14352 = 23 × (54 – 1) 14641 = (1+4+6)4×1 (PF) 14645 = (5 + 6)4 + 4×1 14647 = (4 + 7)4 + 6×1 15003 = 3 × 5001 15030 = 3 × 5010 15125 = 5 × (5 × 11)2 15246 = 6 × 2541 15300 = 3 × 5100 15345 = 3 × 5 × (45 – 1) 15378 = 7 × (5 + 8)3 – 1 15379 = 7 × (5 + 9 – 1)3 15435 = 3 × 5145 15495 = 15 × (45 + 9) 15552 = (15 + 5)5 × 2 15562 = 1 × 2 × (65 + 5) 15567 = 56 – 57 – 1 15568 = 56 – 58 + 1 15585 = 1 × (55 – 8) × 5 15586 = 56 – 5 × 8 + 1 15612 = – 1 + 56 – 12 15613 = 1 + 56 – 13 15615 = 56 – 5 × (1 + 1) 15617 = –1 + 56 – 1 × 7 15618 = 1 + 56 – 1 × 8 15620 = 56 – 10 / 2 (TG) 15621 = –1 + 56 – 2 – 1 15622 = 1 + 56 – 2 – 2 15623 = –1 + 56 + 2 – 3 15624 = 1 + 56 + 2 – 4 15625 = 56 × 125 15626 = 1 + 56×2–6 15627 = 56 + 2 × 17 15628 = 56 + 8 / 2 – 1 15629 = 56 + (9 – 1) / 2 15631 = 56 + (1 + 1) × 3 15632 = 1 + 56 + 3 × 2 15633 = – 1 + 56 + 3 × 3 15634 = 56 + 13 – 4 15635 = 56 + 5 × (3 – 1) 15641 = 56 + 41+1 15642 = 1 + 56 + 42 15645 = 1× 56 + 4 × 5 15655 = 1 × 5 × (6 + 55) 15656 = 1 + 56 + 5 × 6 15661 = 56 + 61+1 15662 = 1 + 56 + 62 15667 = 1 × 56 + 6 × 7 15679 = 56 + 9 × (7 – 1) 15688 = –1 + 56 + 8 × 8 15689 = 56 + 8 × (9 – 1) 15697 = 56 + 9 × (7 + 1) 15698 = 1 + 56 + 9 × 8 15795 = 9 × 1755 15975 = 5 × 5 × 9 × 71 16225 = 6 × 522 + 1 16245 = 5 × (61 – 4)2 16272 = 6 × 2712 16295 = (1 + 6)5 – 29 16347 = 47 – 36 – 1 16348 = 48–1 – 36 16368 = 8 × 31 × 66 16372 = (1 + 3)7 – 6 × 2 16374 = 47 – 1 – 6 – 3 16375 = (5 – 1)7 – 6 – 3 16377 = (1 + 6 – 3)7 – 7 16378 = (8 – 3 – 1)7 – 6 16381 = (1 + 1)6+8 – 3 16382 = (3 – 1)6+8 – 2 16384 = 163 × (8 – 4) (TG) 16385 = (5 – 3)6+8 + 1 16387 = (1 – 6/8)-7 + 3 (JD) 16447 = – 1 + 64 + 47 16448 = 48–1 + 64 16479 = 47 + 96 – 1 16743 = 76–1 – 43 16758 = 75 – 6 × 8 – 1 16759 = 75 – 6 × (9 – 1) 16765 = 75 – 6 × (6+1) 16783 = 76–1 – 8 × 3 16794 = 76–1 – 9 – 4 16797 = 76 / 7 – 9 – 1 16798 = 76 / (8 – 1) – 9 16807 = 76–1 + 0 × 8 16815 = (1 + 6)5 + 1 × 8 16875 = 1 × 68 + 75 16879 = 76–1 + 8 × 9 17253 = (72 – 1) × 35 17325 = 75 × 231 17328 = 8 × (37 – 21) 17346 = 6 × 7 × 413 17368 = 8 × (37 – 16) 17384 = 8 × (37 – 14) 17428 = 2 × 8714 17437 = 47 × 371 17482 = 2 × 8741 17488 = 8 × (4 – 1)7 – 8 17536 = 1 + 75 + 36 17689 = (7 × 19)8–6 17856 = 8 × (56 – 1) / 7 17892 = 9 × 28 × 71 17920 = 70 × 29–1 17925 = 5 × (7 × 29 + 1) 18225 = 81 × 225 18265 = 65 × 281 18270 = 21 × 870 18432 = 18 × 43+2 (MR) 18435 = 18 × 45 + 3 18522 = 2 × 218–5 18594 = 18 × (45 + 9) 18723 = 3 × (71 + 8)2 18744 = 71 × (44 + 8) 19026 = 21 × 906 19215 = 21 × 915 19321 = 1 × 1392 19392 = 39 – 291 19453 = 19 × 45 – 3 (MR) 19592 = (5 – 2)9 – 91 19629 = (1 + 2)9 – 6 × 9 19642 = (6 / 2)9 – 41 19653 = 39 – 1 × 5 × 6 19682 = (6 / 2)9 – 18 19683 = 1 × (9 – 6)8 × 3 19684 = (6 × 4 / 8)9 + 1 19692 = (6 / 2)9 + 9 × 1 19693 = (6 – 3)9 + 1 + 9 19732 = 39 + 1 × 72 19734 = 3 × (94 + 17) 19736 = 9 × (37 + 6) – 1 19737 = 9 × (37 + 7 – 1) 19738 = 39 + 7 × 8 – 1 19739 = (–1 + 9) × 7 + 39 19773 = 9 × (7 + 7 – 1)3 19845 = 5 × 49 × 81 20485 = 5 × (20 + 84) 20736 = (2 × 6)7–3 + 0 21175 = 7 × (5 × 11)2 21375 = 3 × 7125 21495 = 21 × 45 – 9 21504 = 21 × 45 + 0 21586 = 86 × 251 21606 = 6 × (602 + 1) 21753 = 3 × 7251 21843 = (48 – 1) / 3 – 2 21844 = (48 – 4) / (1 + 2) 21845 = (48 – 1) / (5 – 2) 21848 = (48 + 8) / (1 + 2) 21870 = 27 × 810 21875 = 7 × (8 – 2 – 1)5 21943 = (2 × 14)3 – 9 21952 = (29 – 1)5–2 (TG) 21953 = (2 × (5 + 9))3 + 1 22264 = 46 × 222 22528 = 22 × (8 / 2)5 22757 = 7 × (572 + 2) 23326 = 3 × 62+3 – 2 23328 = (2 × 33)2 × 8 23392 = 32 × (93 + 2) 23456 = 25 × (36 + 4) 23490 = 290 × 34 23546 = 23 × 45 – 6 23548 = (3 + 4) × 582 23552 = 23 × 25+5 23796 = 6 × (632 – 3) 24336 = (4 × (36 + 3))2 24339 = (4 × 39)2 + 3 24367 = 7 × (63 – 4)2 24375 = (37 + 2) × 54 24385 = (58 / 2)3 – 4 24389 = (2 × 8 + 9 + 4)3 24390 = 293 + 40 24393 = (3 × 9 + 2)3 + 4 24546 = (2 + 4) × (–5 + 46) 24564 = 6 × (4 × 45 – 2) 24566 = 6 × 46 – 2 × 5 24576 = (2 / 4)–5–7 × 6 24584 = 24 × 45 + 8 24586 = 6 × 84 + 2 × 5 24768 = 4 × 72 × 86 24964 = (94 + 64)2 24972 = 4 × (792 + 2) 25105 = 5 × 5021 25137 = 513 × 72 25314 = (154 + 3) / 2 25375 = 35 × 725 25474 = 47 × 542 25510 = 5 × 5102 25725 = 525 × 72 25872 = 528 × 72 25895 = 5 × ((8 × 9)2 – 5) 25921 = (159 + 2)2 26238 = 2 × 2 × 38 – 6 26244 = (2 / 6)–2×4 × 4 (MR) 26348 = 4 × (38 + 26) 26364 = 263 × 6 / 4 26496 = 9 × 46 × 26 26624 = 26 × 24+6 26754 = 546 × 72 26896 = (96 + 68)2 26973 = 37 × 96/2 27436 = (6 × 7 – 4)3 / 2 27634 = 2 × ((6 × 4)3 – 7) 27639 = 27 × 63 – 9 27648 = (7 – 2 / 8) × 46 27653 = 63 × 27 + 5 27654 = 27 × 45 + 6 27783 = (3 × 7)8/2 / 7 27889 = (79 + 88)2 28217 = (21 × 7)2 – 7 28224 = (2 + 82)2 × 4 28226 = (28 × 6)2 + 2 28322 = 2382 / 2 28476 = 7 × (46 – 28) 28547 = (8 + 5)4 – 7 × 2 28554 = (8 + 5)4 – 5 – 2 (TG) 28556 = (8 + 5)6–2 – 5 28559 = –2 + (8 + 5)–5+9 28561 = 1 × (8 + 5)6–2 (TG) 28564 = (8 + 5)4 + 6 / 2 (TG) 28671 = (2 / 8)–6 × 7 – 1 28672 = 7 × (8 – 2 – 2)6 28674 = 7 × (8 – 4)6 + 2 28678 = 7 × (8 × 8)2 + 6 28728 = 7 × (82×2 + 8) 28749 = 7 × (84 + 9 + 2) 28764 = 6 × (2 × 74 – 8) 28784 = 7 × (84 + 16) 28900 = (80 + 90)2 (TG) 29160 = 10 × (6 × 9)2 29184 = 4 × 8 × 912 29282 = 2 × (9+2)8/2 (MR) 29517 = (95 – 1) / 2 – 7 29519 = (95 – 9) / 2 – 1 29523 = 95 / 2 – 3 / 2 29524 = (2 × 95 – 2) / 4 29525 = (95 – 5) / 2 – 2 29526 = (95 + 6 / 2) / 2 29527 = (95 + 7 – 2) / 2 29529 = 95 / 2 + 9 / 2 29531 = (95 + 13) / 2 29549 = (95 + 49) / 2 29584 = (4 × 5 × 9 – 8)2 29632 = 32 × 926 29768 = 8 × (9 × 6 + 7)2 29795 = 59 × (29 – 7) 29929 = (9 × 9 + 92)2 30625 = (3 × 60 – 5)2 31250 = 10 × (2 + 3)5 31252 = 2 × (52×3 + 1) 31256 = 1 × 2 × (56 + 3) 31346 = 43 × 36 – 1 31347 = 43 × 37–1 31509 = 9 × 3501 31590 = 9 × 3510 31682 = 62 × (83 – 1) 32685 = (6 + 2)5 – 83 32697 = 63 × (29 + 7) 32744 = 2 × (47 – 3 × 4) 32747 = 2 × 47 – 3 × 7 32751 = 2 3×5 – 17 32759 = (3 – 2 + 7)5 – 9 32761 = 23(6–1) – 7 (TG) 32762 = 23(7–2) – 6 (TG) 32764 = 23×7–6 – 4 32765 = –3 + (2 × 7 – 6)5 (TG) 32768 = (3 – 2 + 7)6 / 8 32771 = 3 + 27+7+1 (MR) 32772 = 2 × ((7 – 3)7 + 2) 32775 = (7 + 3 – 2)5 + 7 32778 = 27+8 + 7 + 3 32781 = 27+8 + 13 32782 = 83+2 + 7 × 2 32783 = 323 + 7 + 8 32785 = 3 + 2 × 7 + 85 32786 = 27+8 + 3 × 6 32795 = 5 × (97–3 – 2) 32805 = 5 × (38 + 2 × 0) 32815 = 5 × (38 + 2 × 1) 32825 = 5 × (38 + 2 × 2) 32832 = 323 + 82 32835 = 5 × (38 + 2 × 3) 32836 = 323 + 68 32845 = 5 × (38 + 2 × 4) 32849 = 323 + 92 32851 = 215 + 83 32853 = 323 + 85 32854 = 85 + 43 × 2 32855 = 5 × (38 + 2 × 5) 32859 = 85 + 93 – 2 32865 = 5 × (38 + 2 × 6) 32875 = 5 × (38 + 2 × 7) 32885 = 5 × (38 + 2 × 8) 32895 = 5 × (38 + 2 × 9) 33495 = 33 × (45 – 9) 33579 = 7 × 9 × 533 33655 = 53 × 635 33696 = 36 × 936 34425 = 34 × 425 34968 = 3 × (9 × 64 – 8) 34986 = 48 × 93 – 6 34991 = (9 + 9)4 / 3 – 1 34992 = 3 × (9 × 2)4 / 9 34993 = ((9 + 9)4 + 3) / 3 34996 = 6 × (9 + 9)3 + 4 35152 = 2 × (5 × 5 + 1)3 35684 = 85 – 4 × 36 35721 = 35 × 7 × 21 35726 = 72 × 36 + 5 35782 = (57 – 38) / 2 35928 = (52 + 8)3 – 9 35932 = (3 × (9 + 2))3 – 5 35933 = 333 + 5 – 9 (TG) 35937 = (35 + 7 – 9)3 35942 = (42 – 9)3 + 5 (TG) 36457 = (7 × 56 – 4) / 3 36549 = 9 × (46 – 35) 36850 = (36 + 8) × 50 36855 = 63 × 585 36864 = (6 + 6 – 3) × 84 36918 = 9 × (83+1 + 6) 37179 = 37 × (1 + 7 + 9) 37187 = 17 × 37 + 8 37249 = (3 × 7 × 9 + 4)2 37449 = (49 – 4 + 3) / 7 37668 = 6 × 73 × 86 37814 = 74 × (83 – 1) 37840 = 8 × 4730 37845 = 87 × 435 37875 = 75 × (83 – 5) 38416 = 148×3/6 (TG) 38424 = (2 × (3 + 4))4 + 8 38427 = (2 × 7)4 + 8 + 3 38637 = (8 × 7 – 3) × 36 38640 = 30 × (–8 + 64) 38856 = (38 – 85) × 6 38912 = 38 × 29+1 39216 = ((9 – 2)6 – 1) / 3 39283 = 39 × 2 – 83 39288 = 8 × ((9 + 8)3 – 2) 39294 = 2 × (39 – 36) 39295 = (52 + 9)3 – 9 39304 = 343 + 0 × 9 (TG) 39313 = (33 + 1)3 + 9 (TG) 39314 = 343 + 1 + 9 39328 = 2 × 39 – 38 39342 = (39 – 3 × 4) × 2 39343 = 39 + 343 (TG) 39356 = 6 × (39 – 5) / 3 39358 = 39 × (–3 + 5) – 8 39362 = 6 × (39 – 2) / 2 39363 = 39 / 3 × 6 – 3 (MR) 39366 = 39 / (–3 + 6) × 6 (MR) 39368 = 6 × (38 + 1 / 3) 39369 = 3 + 93 × 6 × 9 39372 = (3 + 9 × 37) × 2 39382 = ((3 × 9)3 + 8) × 2 39424 = 29 × (34 – 4) 39456 = 6 × (94 + 3 × 5) 39784 = 8 × 4973 39864 = 6 × (94 + 83) 39945 = 39 × 45 + 9 41323 = 43 × 312 41468 = 4 × (8 × 64 – 1) 41472 = 2 × (1 + 4 + 7)4 41665 = 641 × 65 (GR) 42025 = 2054–2 (TG) 42336 = 6 × (34 + 3)2 42875 = (42–7)8–5 (TG) 42898 = 89 × 482 43264 = (63 – 4 – 4)2 43268 = (63 – 8)2 + 4 43375 = 53 × (73 + 4) 43688 = 86 × (83 – 4) 43689 = (49 + 8) / 6 – 3 43691 = 49 / 6 + 1 / 3 43692 = (49 + 23) / 6 43775 = (4 × 37 + 7) × 5 43932 = 3 × ((9 + 2)3 + 3) 44375 = 54 × (43 + 7) 44676 = 6 × 7446 44977 = (7 + 7)4 + 94 45056 = (50 – 6) × 45 45360 = 35 × 64 + 0 45361 = 35 × 64 + 1 45362 = 35 × 64 + 2 45363 = 35 × 64 + 3 45364 = 35 × 64 + 4 45365 = 35 × 64 + 5 45366 = 35 × 64 + 6 45367 = 35 × 64 + 7 45368 = 35 × 64 + 8 45369 = 35 × 64 + 9 45632 = –45 + 63×2 45684 = 54 × 846 45760 = 65 × 704 45864 = 84 × 546 45873 = 7 × 38 – 54 45927 = ((4 + 5) × 9)2 × 7 45947 = 4 × 5 + 94 × 7 45957 = 7 × (94 + 5) – 5 45978 = 7 × (94 + 8) – 5 46256 = 66 – (4 × 5)2 46368 = 36 × (64 – 8) 46556 = 66 – 4 × 5 × 5 (TG) 46593 = 3 × (56 – 94) 46608 = 66 – 48 + 0 (TG) 46613 = 66 – 43 × 1 (TG) 46615 = 6 × 65 – 41 46619 = 66 – 9 × 4 – 1 (TG) 46624 = 66 – 4 × 4 × 2 (TG) 46626 = –4 + 66 – 26 46630 = 4 + 66 – 30 (MR) 46632 = –4 × 6 + 63×2 46633 = 4 + 66 – 33 (TG) 46635 = 6 × (65 – 4) + 3 46637 = 66 – 4 × 3 – 7 (TG) 46640 = 66 – 4 × 4 + 0 (TG) 46641 = 66 – 4 × 4 + 1 (TG) 46642 = 66 – 4 × 4 + 2 (TG) 46643 = 66 – 4 × 4 + 3 (TG) 46644 = 4 + 66 – 4 × 4 46645 = 66 – 4 × 4 + 5 (TG) 46646 = 66 – 4 × 4 + 6 (TG) 46647 = 66 – 4 × 4 + 7 (TG) 46648 = 4 × 66 / 4 – 8 46649 = 66 – 4 × 4 + 9 (TG) 46650 = 66 – 5 – 40 46651 = –4 + 6 × 65 – 1 (TG) 46652 = –4 + (6 × 6)5–2 46653 = 66 – (5 + 4)/3 (TG) 46655 = 4 + 6 × 65 – 5 (TG) 46656 = (–4 × 6 + 6 × 5)6 46657 = 67 / 6 + 5 – 4 (TG) 46658 = 6 × 65 + 8 / 4 (TG) 46660 = 4 + 66 + 6 × 0 (TG) 46661 = 66 + 4 + 16 46662 = 62 × 64 + 6 46663 = 4 + 66 + 6 – 3 (TG) 46664 = 66 + 4 × (6 – 4) (TG) 46665 = 6 × (65 + 6 / 4) 46668 = 66 + 6 × 8 / 4 (TG) 46672 = 67 / 6 + 42 (TG) 46673 = –4 + 66 + 7 × 3 (TG) 46677 = 66 + 4 × 7 – 7 (TG) 46684 = –4 + 66 + 8×4 (TG) 46688 = (4 + 66 / 8) × 8 46691 = 66 + 4 × 9 – 1 (TG) 46851 = (4 – 1) × (56 – 8) 46875 = (4 + 7 – 8) × 56 47538 = 57 × 834 47652 = 76 × (54 + 2) 48672 = 78 × 624 48750 = 78 × 54 + 0 48751 = 78 × 54 + 1 48752 = 78 × 54 + 2 48753 = 78 × 54 + 3 48754 = 78 × 54 + 4 48755 = 78 × 54 + 5 48756 = 78 × 54 + 6 48757 = 78 × 54 + 7 48758 = 78 × 54 + 8 48759 = 78 × 54 + 9 49152 = (4 – 1) × 29+5 49277 = 9 × 742 – 7 49584 = 48 × (45 + 9) 49855 = 59 × 845 49896 = 6 × 84 × 99 49968 = 8 × 9 × 694 51200 = 50 × 210 51398 = (59 – 1) / 38 51759 = 9 × 5751 52168 = 8 × 6521 52429 = (49 + 2 / 2) / 5 52483 = 2 × 4 × 38 – 5 52488 = (5 + 2 – 4)8 × 8 52493 = 23 × 94 + 5 52498 = 8 × 94 + 2 × 5 52731 = 217 × 35 52947 = 49 / 2 – 57 53245 = 52 × 45 – 3 53248 = 52 × 48–3 53297 = 2 × 75 + 39 53824 = (8 × (34 – 5))2 53865 = 63 × 855 54369 = (3 + 4) × (65 – 9) 54378 = 87 × 54 + 3 54432 = (4 + 3) × (4 + 2)5 54436 = (4 + 3) × 65 + 4 54476 = 7 × 65 + 44 54642 = 42 × (64 + 5) 54726 = 7 × (65 + 42) 54768 = 7 × (65 + 48) 54872 = (8 × 5 – 2)7–4 54953 = 95 – (3 + 5)4 54958 = 95 – 84 + 5 55225 = (5 × (52 – 5))2 55296 = 54 × 210 56295 = 9 × 6255 56628 = (5 + 8) × 662 56732 = 26 × (37 – 5) 56875 = 65 × 875 57288 = 8 × (25+8 – 7) 57644 = 4 × (6 × 74 + 5) 57645 = 57 – 5 × 46 58921 = 95 – 28–1 58957 = 95 – 28–1 58971 = 95 – 78 × 1 58973 = 95 – 83 + 7 58978 = 95 – 8 × 8 – 7 59032 = 95 – 20 + 3 59038 = 95 – 8 – 3 – 0 59039 = 95 – 9 – 30 59044 = 95 – 4 – 40 59045 = 95 – 4 – 0 × 5 59046 = 95 – 4 + 60 59048 = 95 – 480 59049 = 95 + 0 × 4 × 9 59050 = 95 + 50 + 0 59051 = 95 + 50 + 1 59052 = 5 + 90+5 – 2 59053 = 95 + 50 + 3 59054 = 95 + 50 + 4 59055 = 95 + 50 + 5 59056 = 95 + 50 + 6 59057 = 95 + 50 + 7 59058 = 95 + 50 + 8 59059 = 95 + 50 + 9 59064 = 95 + 60 / 4 59094 = 9 × (94 + 1) + 0 59128 = 95 + 81 – 2 59129 = 95 + 92 – 1 59147 = 95 + 7 × 14 59263 = 95 + 63 – 2 59265 = 95 + 65–2 59273 = 95 + 7 × 32 59313 = 393 – 5 – 1 (TG) 59314 = 394–1 – 5 (TG) 59318 = 398–5 – 1 (TG) 59319 = 399–5–1 (TG) 59375 = (9 + 3 + 7) × 55 59392 = 95 + (9 – 2)3 59409 = 95 + 4 × 90 59451 = 19 × (55 + 4) 59759 = ((5 + 9)5 + 7) / 9 61435 = 5 × (3 × 64 – 1) 61440 = 60 × 44+1 62476 = 6 × ((7 – 2)6 – 4) 62503 = (503 + 6) / 2 62504 = 4 × (56 + 20) 62564 = 4 × 56 + 26 62968 = 68 × 926 63478 = 48 – 6 73 63895 = 65 × 983 63904 = 403 – 96 63945 = 63 × (–9 + 45) 63985 = (8 × 5)3 – 6 – 9 63994 = (49 – 9)3 – 6 64036 = 403 + 6 × 6 (TG) 64512 = 45 × (26 – 1) 64513 = 63 × 45 + 1 64522 = 2542 + 6 (TG) 64550 = (64 – 5) × 50 64868 = 48 – 668 65344 = 64 × (45 – 3) 65471 = –65 + 47+1 65478 = 48 – 65 + 7 65480 = 48 – 56 + 0 65481 = 48 – 56 + 1 65482 = 48 – 56 + 2 65483 = 48 – 56 + 3 65484 = 48 – 56 + 4 65485 = 48 – 56 + 5 65486 = 48 – 56 + 6 65487 = 48 – 56 + 7 65488 = 48 – 56 + 8 65489 = 48 – 56 + 9 65491 = 169–5 – 45 65528 = 25+5+6 – 8 (TG) 65531 = (5 – 3)16 – 5 (TG) 65536 = (6 / 3)6+5+5 (TG) 65542 = 45+5–2 + 6 65841 = 48 + 5 × 61 65884 = 48 + 6 × 58 66339 = (6 × 6)3 + 39 66554 = 65 × 45 – 6 67149 = 9 × 7461 67228 = 28 × 76–2 67234 = 6 + 72+3 × 4 67252 = 2 × 2 × (75 + 6) 67254 = 4 × (75 + 6) + 2 67392 = 72 × 936 67950 = 75 × 906 68644 = (44 + 6)8–6 68800 = 8 × 8600 69253 = 95 × 36 – 2 69255 = 95 × (5 – 2)6 69472 = 67 / 4 – 29 69822 = 862 × 92 69895 = 9 × (65 – 9) – 8 69975 = 67 / (9 – 5) – 9 69984 = 6–9/9+8 / 4 69985 = 9 × 65 + 9 – 8 69993 = 96 × 93 + 9 70225 = (270 – 5)2 71199 = 9 × 7911 72576 = 567 × 27 73125 = 13 × 752 73926 = 6 × 9 × 372 73984 = (8 × 34)9–7 74183 = 31 × (74 – 8) 74353 = 34 × 37 – 5 74358 = 34 × (8 – 5)7 74533 = 73 × (45 – 3) 74536 = 56 × (4 + 7)3 74892 = (4 + 8) × 792 74897 = (87 / 4 + 9) / 7 75433 = 47 + (3 × 3)5 76335 = 35 × (37 – 6) 76832 = 2 × (6 + 8)7–3 76835 = (6 + 8)5 / 7 + 3 77459 = 57 – 9 × 74 78055 = 58 / 5 – 70 78115 = 57 – 8 – 1 – 1 78116 = (6 – 1)7 – 8 – 1 78117 = (6 – 1)7 – 8 × 1 78123 = (8 – 3)7 – 2 × 1 78125 = 57 × 182 78126 = (8 – 6 / 2)7 + 1 78132 = (2 + 3)7 + 8 – 1 78133 = (3 + 3 – 1)7 + 8 78135 = 57 + 8 + 3 – 1 78136 = (6 – 1)7 + 8 + 3 78152 = 57 + 28 – 1 78163 = (6 – 1)7 + 38 78165 = 57 + 8 × (6 – 1) 78225 = 57 + (2 + 8)2 78545 = 57 + 5 × 84 78605 = 57 + 6 × 80 78659 = 57 + 6 × 89 78732 = (7 + 7 – 2) × 38 78975 = 9 × 8775 79299 = 92 × 979 81225 = 1 × 2852 (TG) 81648 = (8 × 8 – 1) × 64 81920 = 80 × 29+1 82372 = 2872 + 3 82755 = 5 × (75 – 28) 82936 = (3 × 96)2 – 8 82942 = (4 × 8 × 9)2 – 2 82944 = (9 × 44 / 8)2 82952 = (9 × 25)2 + 8 83357 = 73 × 25 + 8 83521 = (25 – 8)3+1 (TG) 83524 = (25 – 8)4 + 3 (TG) 83957 = 57 + 8 × 93 84375 = 5 × (7 + 8)4 / 3 84672 = 48 × (6 × 7)2 85264 = (4 × (68 + 5))2 85293 = (2 × 9 – 5) × 38 85358 = (5 + 8) × (38 + 5) 86142 = 21 × (84 + 6) 86724 = (74 + 8) × 62 87381 = (87 / 8 – 1) / 3 87382 = (87 / 8 + 2) / 3 91125 = (9 × 5 × 1)2+1 91853 = (9 + 5) × 38 – 1 91854 = (9 + 5) × (4 – 1)8 92160 = 10 × 962 93184 = 91 × 48–3 93217 = 97 × 312 93294 = 2 × ((4 × 9)3 – 9) 93312 = 2 × (9 × (3 + 1))3 93642 = (9 × 34)2 + 6 94395 = 93 × (45 – 9) 95232 = 93 × 22×5 95234 = 93 × 45 + 2 97333 = (39 + 7)3 – 3 97336 = (39 + 7)6–3 97343 = (49 – 3)3 + 7 97375 = 779 × 53 97966 = 76 – (9 – 6)9 98256 = 6 × (29+5 – 8) 98304 = 3 × 89–4 + 0 98305 = 3 × 85 + 90 98325 = 3 × (85 + 9 – 2) 98375 = 5 × (9 × 37 – 8) 98415 = 98–4 × 15 (MR) 98435 = 5 × (39 + 8 – 4) 99225 = ((9 – 2) × 9 × 5)2

In 2013, the 6–digit Friedman numbers were computed by Giovanni Resta! They can be found here.

Here are the known 6–digit Friedman numbers whose digits are in the correct order:

 103823 = (–1 + 0 + 3×8×2)3 114244 = (1 + 14 – 2)4 × 4 116565 = (–1 + 16) × (–5 + 65) 117128 = 117–1–2 × 8 117476 = 1 – 174 + 76(Jean Marc Falcoz) 117576 = 1 + 1 –75 + 76(Jean Marc Falcoz) 117597 = 11 + (75 – 9) × 7 117619 = –11 + 76 – 19 117624 = –1 × 1 + 76 – 24 117625 = 1 × 1 + 76 – 25 117629 = –1 – 1 + 76 – 2 × 9 117630 = 11 + 76 – 30 117633 = 11 + 76 – 33 117635 = 1 × 1 + 76 – 3 × 5 117637 = –1 – 1 + 76 – 3 – 7 117638 = 1 – 1 + 76 – 3 – 8 117639 = 1 + 1 + 76 – 3 – 9 117641 = –11 + 76 + 4 – 1 117642 = –1 × 1 + 76 – 4 – 2 117643 = 1 × 1 + 76 – 4 – 3 117644 = –1 + (–1 + 76 / 4) × 4 117646 = –1 × 1 + 76 + 4 – 6 117647 = 1 × 1 + 76 + 4 – 7 117648 = 11 + 76 – 4 – 8 117650 = 1 × 1 + 76 + 5 × 0 117651 = –1 – 1 + 76 + 5 – 1 117652 = 1 – 1 + 76 + 5 – 2 117653 = 1 + 1 + 76 + 5 – 3 117655 = 1 + (1 + 76 / 5) × 5 117660 = 11 + 76 + 6 × 0 117662 = 1 × 1 + 76 + 6 × 2 117663 = 11 + 76 + 6 – 3 117695 = 1 × 1 + 76 + 9 × 5 117763 = 117 + 76 – 3 117777 = (1 + 1)7 + 77 / 7 118328 = (1 + (–1 + 8)3)2 – 8 124386 = (124 + 3 – 8) × 6 124416 = ((1 + 2) × 4)4) × 1 × 6 125003 = 1 + 2 + (50 + 0)3 125012 = 12 + 501+2 128500 = (1 + 28) × 500 129283 = (–1 + 29) × (28 – 3) 131071 = (–1 + 3)10+7 – 1 131072 = (1 + 3)1+0+7 × 2 134456 = (1 × 3 + 4)4 × 56 136162 = 1 + (3 + 61 × 6)2 137718 = (–1 + 37) × 7 × (1 + 8) 137772 = (–1 + 37 × 7) × (7 + 2) 137781 = 1 × 37 × 7 × (8 + 1) 137790 = (1 + 37 × 7) × 9 + 0 137791 = (1 + 37 × 7) × 9 + 1 137792 = (1 + 37 × 7) × 9 + 2 137793 = (1 + 37 × 7) × 9 + 3 137794 = (1 + 37 × 7) × 9 + 4 137795 = (1 + 37 × 7) × 9 + 5 137796 = (1 + 37 × 7) × 9 + 6 137797 = (1 + 37 × 7) × 9 + 7 137798 = (1 + 37 × 7) × 9 + 8 137799 = (1 + 37 × 7) × 9 + 9 137839 = –1 + 3 + 7 × (8 + 39) 137948 = –1 + 3 × 7 × (94 + 8) 139965 = –1 × 3 + (9 + 9) × 65 139966 = 1 + 3 × (–9 / 9 + 66) 146410 = (1 + 4 + 6)4 × 10 146461 = (14 + 6)4 × 61 146875 = (1 + 4)6 × (8 + 7 / 5) 147249 = (1 + 47 – 24) × 9 147349 = 1 + (47 – 3 × 4) × 9 147419 = –1 + (47 – 4) × 1 × 9 147429 = (–1 + 47) – 4 + 2) × 9 147447 = (–1 + 47) × (4 × 4 – 7) 147453 = 1 × 47 × (4 + 5) – 3 147455 = –1 + 47 × 45 / 5 147491 = (1 × 47 + 4) × 9 – 1 147519 = (1 + 47 + 5 + 1) × 9 151875 = 1 / 5 × 1 × (8 + 7)5 155520 = (15 + 5)5 × 20 155850 = 1 × (55 – 8) × 50 156225 = (–1 + (56 – 2) × 2) × 5 156235 = 1 × (56 × 2 – 3) × 5 156245 = (–1 + (56 × (–2 + 4)) × 5 156249 = –1 + 56 × 2 × (–4 + 9) 156250 = 1 × 56 × (2 + 5) + 0 156251 = 1 × 56 × (2 + 5) + 1 156252 = 1 × 56 × (2 + 5) + 2 156253 = 1 × 56 × (2 + 5) + 3 156254 = 1 × 56 × (2 + 5) + 4 156255 = 1 × 56 × (2 + 5) + 5 156256 = 1 × 56 × (2 + 5) + 6 156257 = 1 × 56 × (2 + 5) + 7 156258 = 1 × 56 × (2 + 5) + 8 156259 = 1 × 56 × (2 + 5) + 9 156275 = ((–1 + 56) × 2 + 7) × 5 156285 = ((–1 + 56 × 2 + 8) × 5 156295 = 1 × (56 × 2 + 9) × 5 157463 = –1 + ((5 + 7) × 4 + 6)3 158466 = (15 – 8)4 × 66 161051 = (16 + 10)5 × 1 163855 = (–1 + 6) × 3 + 85 × 5 163875 = (163 × 8 + 7) × 5 167042 = (16 + 70)4 × 2 167286 = (1672 – 8) × 6 170471 = (1 × 7 + 0)4 × 71 175232 = (–1 + 75)2 × 32 175274 = 1 + (75 – 2) × 74 176466 = (–1 + 76) / 4 × 6 – 6 176469 = (1 + 76 – 4) / 6 × 9 176472 = (–1 + 76) × (–4 + 7) / 2 177147 = (1 + 7/7 + 1)4+7 182476 = (1 + 8 – 2)4 × 76 184275 = (–1 + 84) × (2 + 7) × 5 184325 = (1 + 84 × 32) × 5 184329 = (1 + 84 × (3 + 2)) × 9 184335 = (1 + 84 × 3) × 3 × 5 184365 = (1 + 84) × (3 + 6) × 5 184495 = (–1 + (84 + 4) × 9 ) × 5 184545 = 1 × (84 + 5) × 45 185193 = 1 × ((8 – 5) × 19)3 186615 = –1 + 8 + 66 × (–1 + 5) 186622 = 1 × 8 × 66 / 2 – 2 186624 = (18 × (6 + 6))2 × 4 186628 = (18 + 66) / 2 × 8 186631 = –1 + 8 + 66 × (3 + 1) 186641 = 18 + 66 × 4 – 1 186642 = (1 + 8) × ((6 + 6)4 + 2) 186644 = (18 + 66 + 4) × 4 186646 = (–1 + 8 + 66) × 4 – 6 186648 = (1 × 8 + 66) × 4 – 8 186684 = (–1 + 8 + 66 + 8) × 4 187278 = ((–1 + 8) × 7)2 × 78 196608 = ((–1 + 9)6 × 6 + 0) / 8 196830 = 1 × (9 – 6)8 × 30 206486 = (20 + 6)4 × 86 209944 = (2 + 0) × ((9 + 9)4 – 4) 209946 = (2 + 0) × (9 + 9)4 – 6 209952 = (2 × (0 + 9))9–5 × 2 210125 = (210 + 1)2 / 5 216003 = 2 + 1 + (60 + 0)3 216021 = 21 + 602+1 218491 = (–2 + 1 + 8)4 × 91 227529 = (22 × 7 + 5)2 × 9 229373 = 22×9–3 × 7 – 3 229378 = 2 + 29+3 × 7 × 8 232324 = (–2 + 32+3)2 × 4 233255 = ((2 × 3)3×2 – 5) × 5 233280 = (2 × 33)2 × 80 234224 = –2 + 34 + 224 234248 = ((2 + 3) × 4 + 2)4 – 8 234254 = –2 + (34 / 2 + 5)4 234264 = 23 + (–4 + 26)4 234375 = (2 + 3)4 × 375 234377 = 2 + 3 × (4 × 3 – 7)7 235296 = –2 + (–3 + 5) × (–2 + 9)6 235768 = 2 × (35 + 76 – 8) 236194 = –2 + 36 × 1 × 94 236196 = 2 × 36+1 × 9 × 6 236764 = 2 × (36 + 76 + 4) 238648 = 23 × (8 × 64 + 8) 245760 = (–2 + 4)5+7 × 60 247167 = 2 × 47 – 1 + 67 248830 = –2 + (4 + 8)8–3 + 0 248831 = –2 + (4 + 8)8–3 + 1 248832 = –2 + (4 + 8)8–3 + 2 248833 = –2 + (4 + 8)8–3 + 3 248834 = –2 + (4 + 8)8–3 + 4 248835 = –2 + (4 + 8)8–3 + 5 248836 = –2 + (4 + 8)8–3 + 6 248837 = –2 + (4 + 8)8–3 + 7 248838 = –2 + (4 + 8)8–3 + 8 248839 = –2 + (4 + 8)8–3 + 9 249318 = (2 + 4 × 9) × (3 × 1)8 250002 = 2 + (500 + 0)2 253135 = (2 + (5 + 3)1+3) × 5 255886 = –2 × 55 – 8 + 86 257049 = (25 + 7 + 0)4 / 9 259549 = –2595 + 49 261883 = –261 + (8 × 8)3 262118 = –26 + (2 × 1)18 262122 = 26×(2+1) – 22 262128 = 26×(2+1) – 2 × 8 262136 = –2 + 6 + (2 × 1)3×6 262137 = (–2 + 6)2+1+3 – 7 262139 = –2 + 6 / 2 + (1 + 3)9 262140 = 26×(2+1) – 4 + 0 262141 = 26×(2+1) – 4 + 1 262142 = 26×(2+1) – 4 + 2 262143 = 26×(2+1) – 4 + 3 262144 = 26×(2+1) – 4 + 4 262145 = 26×(2+1) – 4 + 5 262146 = 26×(2+1) – 4 + 6 262147 = 26×(2+1) – 4 + 7 262148 = 26×(2+1) – 4 + 8 262149 = 26×(2+1) – 4 + 9 262156 = 2 × 6 + (2 + 1 + 5)6 262176 = 26 / 2 + (1 + 7)6 262196 = 26 × 2 + (–1 + 9)6 262286 = (2 × 6)2 – 2 + 86 262438 = –2 + (62 + 4) × 38 262440 = (2 / 6)–2×4 × 40 263866 = (2 × 6)3 + 86 – 6 263866 = –2 – 6 + 56 × 17 265689 = 26 + 56 × (8 + 9) 266565 = 26+6 + 5 × 65 268321 = –2 + (6 + 83)2 – 1 268323 = 2 + (6 + 83)2 – 3 268324 = (2 × (6 + 83))2 / 4 273375 = (2 + 7)3) × 375 274673 = –2 + (7 – 4 + 62)3 275686 = (2 × 7)5 + 6 – 86 279666 = ((2 – 7) × 9 + 66) × 6 279841 = (2 × 7 + 9)8–4×1 279867 = 2 – 79 + 8 + 67 279934 = –2 + (7 – 9 / 9)3+4 279936 = ((2 – 7 + 9) × 9)3 × 6 279937 = (2 + 7) / 9 + (9 – 3)7 279967 = 279 / 9 + 67 282240 = (2 + 82)2 × 40 287496 = ((2 + 8) × 7 – 4)9–6 289536 = 28 × (9 × 53 + 6) 291602 = 2 + (9 × 1 × 60)2 294778 = 2 × 9 × (47 – 7) – 8 294782 = –2 + 94 × (7 × 8)2 294829 = –2 + 9 × (48 / 2 – 9) 294838 = –2 + 9 × (4 × 8)3 – 8) 294894 = 2 × (–9 + 48 × 9 / 4) 294895 = (2 + (94 – 8) × 9) × 5 294912 = 2 × 9 × 49–1×2 294928 = 2 × (9 × 49–2 + 8) 295195 = (–2 + 95 + 1 – 9) × 5 295225 = (–2 + 95 / 2) × 2 × 5 295235 = (–2 + 95) × (–2 + 3) × 5 295243 = –2 + 95 × (2 × 4 – 3) 295245 = 2 × 95 × 2 / 4 × 5 295247 = 2 + 95 × (2 – 4 + 7) 295255 = (2 + 95) × 25 / 5 295285 = (2 + 95 – 2 + 8) × 5 295465 = (–2 + 95 + 46) × 5 295505 = (2 + 95 + 50) × 5 296346 = ((–2 + 9) × 6)3 × 4 – 6 296384 = ((–2 + 9) × 6)3 + 8 × 4 299575 = (29 + 9) × 575 312325 = 312 × 325 314431 = ((3 + 14) × 4)3 – 1 314928 = (3 × 14)9 × 2 × 8 325125 = ((3 + 2) × 51)2 × 5 326557 = ((3 × 2) × 65 – 5) × 7 326586 = 3 × (–2 + 65 × (8 + 6)) 326592 = 3 × 2 × 65 × (9 – 2)) 326617 = 32 + (66 – 1) × 7 326634 = ((3 × 2)6 + 6) × (3 + 4) 326697 = ((3 × 2)6 + 6 + 9) × 7 327485 = (–32 – 7 + 48) × 5 331683 = 3 × (–31 + (6 × 8)3) 331773 = –3 + (31 – 7)7–3) 333234 = (33 × 3)2 × 34 344250 = 34 × 4250 347736 = 3 + 477 × 36 351232 = (3 + 51 + 2)3 × 2 352926 = 3 × (–5 – 2 + (9 – 2)6) 352932 = 3 × (–5 + (–2 + 9)3×2) 352947 = 3 × (5 + 2)9–4 × 7 352961 = 3 × (5 + (–2 + 9)6) – 1 354277 = ((3 × 5)4 – 2 × 7) × 7 354292 = 35+4 × 2 × 9 – 2 354294 = 35+4 / 2 × 9 × 4 354627 = ((3 × 5)4 + 62) × 7 357210 = 35 × 7 × 210 360855 = (–3 + 6 + 0)8 × 55 367272 = (36 × 7 – 2) × 72 368500 = (36 + 8) × 500 371314 = 3 × 7 + 131+4 372573 = 37 + 2 × 573 373239 = (–3 + 73 + 2)3 – 9 373248 = (3 × (7 – 3))2+4 / 8 374439 = (–3 + 74 × 4) × 39 374529 = 3 × (74 × 52 – 9) 375021 = 3 × (7 + 502+1) 375168 = 3 × (7 + 51×6) × 8 379793 = (–3 + 7)9 + 79–3 386758 = –3867 + 58 388560 = (38 – 85) × 60 388993 = –3 × 8 + (–8 + 9 × 9)3 389342 = ((3 + 89)3 – 4) / 2 390358 = –3 + 90 × 3 + 58 390583 = –39 + 0 + 58 – 3 390589 = (–3 × 9 + 0) + 58 – 9 390628 = 3 + (90 + 6 – 2)8 390658 = (3 × 9 + 0 + 6 + 5)8 391864 = (–39 + (1 – 8)6) × 4 393189 = 3 × (–9 + (3 – 1)8+9) 393216 = (3 + 9 / 3) × 216 393420 = (39 – 3 × 4) × 20 393660 = 39 / (–3 + 6) × 60 393720 = (3 + 9 × 37) × 20 393820 = ((3 × 9)3 + 8) × 20 397535 = (3 × (9 + 7) – 5)3 × 5 344250 = 34 × 4250 357210 = 35 × 7 × 210 368500 = (36 + 8) × 500 388560 = (38 – 85) × 60 390658 = 39 + 0 – 6 + 58 393420 = (39 – 3 × 4) × 20 393660 = 39 / (–3 + 6) × 60 393720 = (3 + 9 × 37) × 20 393820 = ((3 × 9)3 + 8) × 20 411772 = (41–1 + 77) / 2 413466 = (413 – 4 – 6) × 6 413496 = (413 + 4 – 9) × 6 413518 = 413 × (5 + 1) – 8 413536 = (413 + 5 / 3) × 6 417625 = (4 + 176–2) × 5 419904 = 4 × (9 + 9 + 0)4 420175 = (4 + 20 + 1) × 75 425984 = (4 × 2)5 × (9 + 8 – 4) 432964 = 4 × 3296–4 437564 = 43 + 7 × 56 × 4 437656 = 4 × (–3 + 7 × (6 + 56)) 437750 = (4 × 37 + 7) × 50 455625 = (4 × 5 – 5)6 / 25 456533 = (4 + 5 + 65 + 3)3 456976 = (4 × 5 + 6)–9+7+6 459270 = (4 + 5) × 9)2 × 70 466520 = (–4 + 66) × 5 × 2 + 0 466521 = (–4 + 66) × 5 × 2 + 1 466522 = (–4 + 66) × 5 × 2 + 2 466523 = (–4 + 66) × 5 × 2 + 3 466524 = (–4 + 66) × 5 × 2 + 4 466525 = (–4 + 66) × 5 × 2 + 5 466526 = (–4 + 66) × 5 × 2 + 6 466527 = (–4 + 66) × 5 × 2 + 7 466528 = (–4 + 66) × 5 × 2 + 8 466529 = (–4 + 66) × 5 × 2 + 9 466536 = (–4 + (66 × 5 / 3) × 6 466552 = (–4 + 6 × 65 × 5) × 2 466553 = (–4 + 66 × (5 + 5) – 3 466557 = (4 + 66 × (5 + 5) – 7 466560 = (4 + 6) × 65 × 6 + 0 466561 = (4 + 6) × 65 × 6 + 1 466562 = (4 + 6) × 65 × 6 + 2 466563 = (4 + 6) × 65 × 6 + 3 466564 = (4 + 6) × 65 × 6 + 4 466565 = (4 + 6) × 65 × 6 + 5 466566 = (4 + 6) × 65 × 6 + 6 466567 = (4 + 6) × 65 × 6 + 7 466568 = (4 + 6) × 65 × 6 + 8 466569 = (4 + 6) × 65 × 6 + 9 466652 = (46 + 66 × 5) × 2 466880 = (4 + 66 / 8) × 80 470576 = 4 × (7 × 0 – 5 + 76) 470596 = 4 × (7 + 0 × 59)6 470616 = 4 × ((7 + 0)6 – 1 + 6) 470628 = (4 + ((7 + 0)6 / 2)) × 8 470632 = 4 × ((7 + 0)6 + 32) 471576 = (471 + 57) × 6 472364 = 4 × (–7 + 2 × 36+4) 472384 = –4 + 72 × 38 – 4) 472388 = 4 + 72 × 38 – 8) 472392 = (4 + 7 × 2)3 × 92 472395 = –4 + 7 + 23 × 92 472398 = (–4 + 7) × (2 + 39 × 8) 472439 = 47 + 24 × 39 474552 = (4 + 74)5/5+2 475136 = 47 × (5 + (1 + 3) × 6) 475281 = (47 + 5) × (28 + 1) 476254 = 4 + 762 × 54 483153 = ((–4 + 8) × 3 – 1)5 × 3 484128 = (48 × 41)2 / 8 492205 = 492 × 205 493837 = –4 + 9 × 383 – 7 493852 = 4 + 9 × 385–2 497657 = 4 × (9 + 7) × 65 – 2 497662 = (–4 + 9 +7)6 / 6 – 2 497664 = 4 × (9 + 7) × 6 × 64 508276 = (5 + 0)8 + 2 + 76 523665 = ((5 – 2) × 3)6 – 65 524248 = (–5 + (–2 + 4)24) × 8 524268 = (–5 / 2 + 42+6) × 8 524282 = –5 + (–2 + 42+8) / 2 524283 = –5 + 2(4–2)×8+3 524285 = –5 + 2 + 42 × 85 524288 = (5 + (2 – 4) / 2)8 × 8 524293 = 5 + 2 × 4 × 29–3 524298 = 5 × 2 + 4 × 29+8 524880 = (5 + 2 – 4)8 × 80 526833 = (–5 + (26 – 8)3) × 3 531296 = –5 × (31 – 2) + 96 531396 = –5 × 3 × 1 × 3 + 96 531426 = –5 × 3 + (1 + 4 × 2)6 531428 = –5 + 314–2 – 8 531433 = –5 + 31×4×3 – 3 531436 = –5 + 314 / (3 + 6) 531438 = 5 + 31×4×3 – 8 531439 = –5 + 3 + 1 – 43+9 531441 = (5 – 3 + 1)4(4–1) 531443 = 5 + (31– 4)4 – 3 531446 = 5 + 314+4–6 531456 = 5 × 3 × 1 + (4 + 5)6 531494 = 53 + ((–1 + 4) × 9)4 531496 = 5 × (–3 + 14) + 96 531966 = 531 + 96 – 6 538412 = (5 + 38) × 41 × 2 546875 = 54–6+8 × 7 × 5 549365 = 5 × (493 – 65) 551343 = –5 × 5 + (1 + 34)3 559539 = (55 + 95 – 3) × 9 562419 = ((56 – 2) × 4 – 1) × 9 563922 = ((56 + 3) × 9)2 × 2 577602 = –5 + 7 + 7602 578125 = 57 × (8 – 1 + 2 / 5) 583443 = (–5 – 8 + 34)4 × 3 583889 = –5 × 8 + 38 × 89 584647 = (5 + 8 / 4 × 6)4 × 7 585640 = 5 × 8 × (5 + 6)4 + 0 585641 = 5 × 8 × (5 + 6)4 + 1 585642 = 5 × 8 × (5 + 6)4 + 2 585643 = 5 × 8 × (5 + 6)4 + 3 585644 = 5 × 8 × (5 + 6)4 + 4 585645 = 5 × 8 × (5 + 6)4 + 5 585646 = 5 × 8 × (5 + 6)4 + 6 585647 = 5 × 8 × (5 + 6)4 + 7 585648 = 5 × 8 × (5 + 6)4 + 8 585649 = 5 × 8 × (5 + 6)4 + 9 588765 = (5 + 8) × 8 + 76 × 5 589748 = –5 + (–8 + 9 × (–7 + 48)) 589864 = 5 × 8 + 9 × 86 / 4 590945 = 5 + 90 × (94 + 5) 592763 = 59 + (2 × 7 × 6)3 597878 = 5 + (97 + 8 + 7) / 8 606476 = ((6 + 0)6 – 4) × (7 + 6) 614125 = (6 × 14 + 1)–2+5 624978 = –6 + (2 + (–4 + 9)7) × 8 629844 = 6 × (–2 + (9 × 8 / 4)4) 629848 = (6 / 2)9) × 8 × 4 – 8 635993 = –63 + (5 + 9 × 9)3) 640024 = (6 + 4002) × 4 645500 = (64 – 5) × 500 649495 = (6 – 4 + 9) × (–4 + 95) 649529 = –6 – 4 + 95 × (2 + 9) 653184 = 65 × (3 + 18) × 4 655284 = (65 + 52) × 84 655354 = –6 + 5 × (5 + 3)5 × 4 655935 = (6 × 55 – 9) × 35 656187 = (6 × 56 – 1 – 8) × 7 656244 = –6 + 56 × (–2 + 44) 656250 = 6 × 56 × (2 + 5) + 0 656251 = 6 × 56 × (2 + 5) + 1 656252 = 6 × 56 × (2 + 5) + 2 656253 = 6 × 56 × (2 + 5) + 3 656254 = 6 × 56 × (2 + 5) + 4 656255 = 6 × 56 × (2 + 5) + 5 656256 = 6 × 56 × (2 + 5) + 6 656257 = 6 × 56 × (2 + 5) + 7 656258 = 6 × 56 × (2 + 5) + 8 656259 = 6 × 56 × (2 + 5) + 9 656298 = 6 × (56 × (–2 + 9) + 8) 656373 = 6 × (56 + 3) × 7 – 3 656376 = (6 + 56 – 3) × 7 × 6 656418 = 6 × (56 + 4) × (–1 + 8) 656790 = 6 × (56 × 7 + 90) 656817 = (6 × 56 + 81) × 7 657874 = (–6 + 5 × 7 × 8) × 74 658503 = (((6 + 5) × 8) – 50)3 659685 = (65 – 9 – 6) × 85 663552 = 6 × (6 × (3 + 5) + 55–2) 677328 = 6 × ((–7 + 73)2 – 8) 699875 = (69 / 9 / 8 + 7) × 5 705642 = ((7 – 0)5 – 6) × 42 728993 = –7 + ((2 + 8) × 9) × 99/3 741321 = (7 × 41 × 3)2×1 742572 = ((7 + 4 + 2)5 – 7) × 2 742586 = (7 + 4 + 2)5 × (8 – 6) 756045 = (75 – 6 + 0) × 45 756315 = 75 × (6 – 3) × 15 756325 = (75 × (6 + 3) + 2) × 5 756549 = ((75 + 6) × 5 – 4) × 9 756585 = (75 + 6) × (5 × 8 + 5) 759359 = –7 + (5 × 9 / 3)5 – 9 759375 = (–7 + 59 – 37)5 765392 = 76 × 5 + 39+2 766927 = (7 + 6 × 6 × 9)2 × 7 774137 = 7 × (7 + 41)3 – 7 777922 = ((7 + 7) × 7 × 9)2 – 2 777924 = 7 × 7 × (7 × 9)2 × 4 781257 = 7 + (81 + 2) × 57 786369 = (–7 + 86 / (–3 + 6)) × 9 786385 = (–7 + 86 × 3) – 8 × 5 786393 = (–7 + 86 + 3 – 9) × 3 786396 = (–7 + 86) × 3 – 9 – 6 786411 = (–7 + 86) × (4 – 1) × 1 786413 = –7 + (86 – 4) × 1 × 3 786425 = –7 + 8 × 6 × 42+5 786433 = –7 + 8 + 643 × 3 786439 = 7 + (8 × 6 × 4)3 / 9 786441 = 7 + 86 – 4 × (4 – 1) 786453 = 7 + 86 × (4 + 5) / 3 789264 = 7 × (89 – 2) × 64 805255 = (8 + 0 + 5 – 2)5 × 5 805655 = (80 + (5 + 6)5) × 5 806752 = (8 + 0) × 6 × 75 + 2 823297 = –82 × 3 + (–2 + 9)7 823461 = –82 + (3 + 4)6+1 823527 = –8 × 2 + (–3 + 5 × 2)7 823543 = (8 – 2 × 3 + 5)4+3 824577 = 8 + 2 + 45 + 77 839424 = 8 × (–3 + 94) × 24 839673 = ((–8 + 3) × 9 + 67) × 3 839812 = 8 + (–3 + 9)8 × 1 / 2 839827 = –8 + 3 × (9 + (8 – 2)7) 839867 = 8 + 3 × (9 + 8 + 67) 844277 = (8 + 4)4 – 2 + 77 851942 = (85 – 1) × (9 + 4) × 2 851968 = (8 + 5) × (1 + 9 – 6)8 857383 = 8 + (57 + 38)3 861327 = (86 – 1) × (3 + 2 / 7) 875056 = 8 × 7 × (50 + 56) 875336 = 8 × 7 × (53 + 3 + 6) 884733 = (8 × 84 / 7)3 – 3 884736 = 8 × 84 × (7 × 3 + 6) 886464 = (8 × 86 – 4) × 64 907569 = 9 × (0 + 7)5 × 6 – 9 912247 = (9 + 12 – 2)4 × 7 923314 = –9 × 23 + 314 924385 = (9 – 2)4 × 385 937577 = (9 + 3) × (7 + 57) – 7 944784 = 94 × (4 × 7 + 8) + 4 973944 = ((9 × 7)3 – 94) × 4 979767 = –9 + 7 / (9 – 7) × 67 984150 = 98–4 × 150

Here are the number of Friedman numbers known with n digits:

Number of
Digits
Number of
Friedman Numbers
Number of Nice
Friedman Numbers
100
210
3133
45811
577293
68968528

Mike Reid, Philippe Fondanaiche, and I found that 123456789 = ((86 + 2 × 7)5 – 91) / 34 and 987654321 = (8 × (97 + 6/2)5 + 1) / 34 are Friedman numbers. In 2007, Chris Wardle found another pandigital Friedman number. This inspired us to find many more, shown below. Bruno Curfs found another in 2008, and several more in 2013. There is even a nice pandigital Friedman number known: 268435179 = –268 + 4(3×5 – 17) – 9.

 152843769 = (12368 – 5)4+7–9 (BC) 157326849 = 125437+9–6–8 (BC)

 214356789 = (9 + 7 – 6 + 1)8 – 4 × 523 214356798 = (2+9)8 – 6 × 347 – 15 214356879 = (4+7)8 – 91 × (25 – 6 + 3) 214356897 = (7 + 6 – 2)3+5 – 1984 214356978 = (6+5)8 – 1924 + 3 × 7 214356987 = (7 + 6 – 2)3+5 – 1894 214357689 = (4+7)8 – 2 × 596 × 13 214357698 = (5 × 2 + 4 – 3)8 – 7 × 169 214357869 = (9 + 7 – 6 + 1)8 – 4 × 253 214357896 = (4+7)2+6 – 985 × 13 214357968 = (6+5)8 – 913 × (2 × 4 – 7) 214357986 = (4+7)2+6 – 895 × 13 214358679 = (6+5)8 – 217 + 3 × (9 – 4) 214358697 = (6+5)8 – 174 – 9 – 3 + 2 214358769 = (6+5)8 – 127 + 3 × (9 – 4) 214358796 = (6+5)8 – 7 × 4 × 9 / 3 + 2 × 1 (CW) 214358967 = (6+5)8 – 72 + 3 × (9 – 4) – 1 214358976 = (6+5)8 + 1 × 94 – 3 × 2 + 7 (CW) 214359678 = (6+5)8 + 793 + 4 × 12 214359687 = (4 + 5 + 9 – 7)8 + 13 × 62 214359768 = (5+6)8 +37 × 24 – 19 214359786 = (5+6)8 + 912 – 7 × (4 – 3) 214359867 = (5 × 2 + 4 – 3)7+1 + 986 214359876 = (2+9)8 + (4 + 6)3 – 5 × 17 214365789 = (2 × 4 + 3)8 + 6915 – 7 214365798 = (5 × 2 + 4 – 3)8 + 6917 214365879 = (6+5)8 + 7 × (1 + 9)3 – 4 / 2 214365897 = (6+5)8 + 7 × (1 + 9)3 + 42 214365978 = (6+5)8 + 7124 – 3 × 9 214365987 = (6+5)8 + 78 × 91 + 23 245893761 = (15683 – 2)4+7–9 (BC) 254817369 = 159632 × (8 – 7)4 (BC)

 375468129 = (19375 + 2)4+6–8 (BC) 382945761 = (19572 – 3)4+6–8 (BC) 385297641 = (19634 – 5)2×(8–7) (BC) 387412569 = 92+7 – 3 × 48 × (56 – 1) 387412596 = 94+5 – 7861 – 32 387412659 = 94+5 – 7826 – 3 – 1 387412695 = 92+7 – 65 – 3 × (14 – 8) 387412956 = 93+6 – 7528 – 4 – 1 387412965 = 916–8+1 – 7524 387415269 = 94+5 – 87 × (63 – 2 – 1) 387415296 = 92+7 – 5186 – 3 – 4 387415629 = 91+8 – 3 × 27 × (56 + 4) 387415692 = 98+2–1 – 75 × 64 + 3 387415926 = 92+7 – 53 × 86 – 4 – 1 387415962 = 98–4+5 – 62 × 73 – 1 387416259 = 92×6 – 3 – 5 × 18 × 47 387416295 = 92+7 – 46 – 83 – 15 387416529 = 93+6 – 72 × (58 – 4 +1) 387416592 = 93+6 – 54 × 72 – 1 – 8 387416925 = 91+8 – 54 × (67 + 2 –3) 387416952 = 92+7 – 58 × 61 + 4 – 3 387419256 = 93+6 – 82 × 15 – 7 + 4 387419265 = 93(5–2) – 87 × 14 – 6 387419526 = 94+5 – 37 × 26 – 18 387419562 = 94+5 – 78 × 12 + 6 + 3 387419625 = 94+5 – 867 + 2 + 13 387419652 = (6–3)2×9 – 817 – 4 × 5 (CW) 387421569 = 93+6 + 8 × 15 × (7 + 4 – 2) 387421596 = 92+7 + 85 × 13 + 6 – 4 387421659 = 93+6 + 78 × 5 × (4 + 1 – 2) 387421695 = 92+7 + 86 × 14 + 5 – 3 387421956 = 94+5 + 86 × 17 + 2 + 3 387421965 = 92+7 + 36 × (5 × 8 + 14) 387425169 = 91+8 + 72 × 65 × (4 – 3) 387425196 = 92+7 + 84 × 56 × 1 + 3 387425619 = 91+8 + 57 × 3 × (24 + 6) 387425691 = 98/4 + 7 + 512 × 6 / 3 387425916 = 94+5 + 67 × 81 × (3 – 2) 387425961 = 94+5 + 72 × (83 – 6 – 1) 387426159 = 94+5 + 81 × (76 – 2 × 3) 387426195 = 98+4–3 + 5712 – 6 387426519 = 94+5 + 67 × (81 + 32) 387426591 = 93(5–2) + 71 × 86 – 4 387426915 = 93(8–5) + 6427 – 1 387426951 = 913–4 + 85 × 76 + 2 387429156 = 913–4 + 8672 – 5 387429165 = 92+3+4 + 8675 + 1 387429516 = 91+8 + 26 × 347 + 5 387429561 = 98+4–3 + 12 × 756 387429615 = 97+8–6 + 54 × 132 387429651 = 92+7 + 4581 × 6 / 3

 536874912 = 4 × (89 + (2 × 5)3) + 7 – 6 – 1 (BC) 536874912 = (1 + 7 – 6)29 + 4 × 8 × 53 (BC) 537219684 = 231784–5–6+9 (BC) 547981236 = 1534 – 9 × (8 – 7 + 6 – 2) (BC) 547981263 = 1534 – 2 × (78 – 69) (BC) 594823671 = 296 + 5 × 7 × (8 + 4 – 3 + 1) (BC) 594823761 = 296 + (7 + 3) × (8 × 5 × + 4 × 1) (BC)

 672935481 = 25941–3+6+7–8 (BC)

 714653289 = (26738 – 5)(9–1)/4 (BC)

Bruno Curfs also found several 10-digit pandigital Friedman numbers.

 1026753849 = (30249 – 6)7–5 × 18 (BC) 1237069584 = 351726–4 + 89 × 0 (BC) 1436789025 = 379052 × 1468 (BC) 1532487609 = 391472 + 0 × 568 (BC) 1927385604 = 439027–5 × 168 (BC) 2170348569 = 465872 + 139 × 0 (BC) 2913408576 = 539762 + 0 × 148 (BC) 3015986724 = 549182 + 0 × 367 (BC) 3285697041 = 573216–4 + 0 × 89 (BC) 3412078569 = 584132 + 0 × 679 (BC) 3528716409 = 594032 × 1678 (BC) 3719048256 = 609842 × 1357 (BC) 3975428601 = 630512 + 9 – 7 – 8/4 (BC) 3985270641 = 631297–5 + 0 × 48 (BC) 4832057169 = 695132 + 0 × 478 (BC) 5368709124 = (8 – 6)29 × (3 + 7) + 4 + 15 × 0 (BC) 5803697124 = 761825–3 + 0 × 49 (BC) 6154873209 = 784532 + 0 × 169 (BC) 6457890321 = 803612 × (5 – 4)79 (BC) 6714983025 = 819452 + 0 × 367 (BC) 7351862049 = 857432 + 0 × 169 (BC) 7408561329 = 860732 × 1459 (BC) 7680594321 = 876392 + 0 × 145 (BC) 7946831025 = 891452 + 0 × 367 (BC) 8014367529 = 895236–4 + 0 × 17 (BC) 8127563409 = 901532 × (7 – 6)48 (BC) 8326197504 = 912485–3 + 0 × 67 (BC) 8391476025 = 916052 + 3 + 8 – 4 – 7 (BC) 9351276804 = 967023–1 × (5 – 4)8 (BC)

Bruno Curfs sent these files of pandigital vampire numbers, which are therefore Friedman numbers. Some have 9-digits and others have 10 digits.

Philippe Fondanaiche says the smallest repdigit Friedman number appears to be 99999999. I improved some of his smallest repdigits, which are shown below.

 11111111111 = ((11–1)11 – 1×1) / (11–1–1) 22222222222222 = (2((22–2)/2)22+2–2–2) / (2+2/2)2 333333333 = ((3×3 + 3/3)3×3 – 3/3) / 3 444444444444444 = (4(44/4 – 4/4)4×4–4/4 – 4) / (4 + 4 + 4/4) 5555555555 = (5(5+5)5+5 – 5) / (5 + 5 – 5/5) 6666666666666666 = (6((66–6)/6)6 + (66–6)/6 – 6) / (6 + (6+6+6)/6) 77777777777777 = (7((77–7)/7)7+7 – 7 + 7 – 7) / (7 + (7+7)/7) 88888888888888 = (8((88–8)/8)8+8–(8+8)/8 – 8) / (8 + 8/8) 99999999 = (9 + 9/9)9–9/9 – 9/9

Brendan Owen proved that repdigits of length 24 or more are Friedman numbers in any base, by showing:

aaa...a = (a×a / (aa–a–a) ) ( ( (aa–a)/a)A + (a+a+...+a)/a – a/a), where A = ((a+a+a+a+a)/a)(a+a)/a – a/a.

Robert Reid notes that 7776 would be a Friedman number if we allowed decimal points: 7776 = 67/(.7+.7). Jacon Minninga showed the same about 4875 = 7.8 × 54. Bruno Curfs pointed out that 111111111 = ((1/.1)1/.1–1 – 1)/(1/.1 – 1) and dddddddddd = (d × (d/.d)d/.d – d)/(d/.d – d/d) for any other digit d.

The following people disproved that F(n)/n → 0. Brendan Owen used N,12588304 = N × 108 + 35482 (along with dozens of larger examples). Mike Reid used N,46656 = N × (4+6)5 + 66. Note that these examples also show that there are Friedman numbers beginning with any string of digits.

Michael Brand sent me an amazing proof in March of 2010 that F(n)/n → 1. It was published in 2013: Michael Brand, "Friedman numbers have density 1", Discrete Applied Mathematics, 161 (16-17), pp. 2389-2395, 2013.

Ulrich Schimke conjectures that for every k which is not a power of 10, kn is a Friedman number for arbitrarily large n. He notes that 2n appears to be a Friedman number for all n>9. Trevor Green points out that all powers of 5 seem to be Friedman numbers.

A vampire number is a number that can be written as the product of numbers that together contain the same digits as the number itself. This sequence, a subsequence of the Friedman numbers, begins 126, 153, 688, 1206, 1255, 1260, 1395, 1435, 1503, 1530, 1827, 2187, 3159, 3784, 6880, . . . , and is sequence A020342 of the Encyclopedia of Integer Sequences. Philippe Fondanaiche sent me lots of vampire numbers, but noticed that vampire numbers get more and more rare, so that the vast majority of Friedman numbers use the exponential operator.

A pretty wild narcissistic number uses the digits in order, but also allows the square root and factorial operators. So nice Friedman numbers are a subset of these numbers.

Philippe Fondanaiche noticed that most Friedman numbers are composite. The first few known prime Friedman numbers are 127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 19739, . . . .

Ron Kaminsky proved that there are infinitely many prime Friedman numbers. The numbers k×1014+19683 = k×106+8+39+0+0+ . . . are Friedman numbers for all k. The numbers of this form are an arithmetic sequence a n+b where a and b are relatively prime, and therefore, by a well–known theorem of Dirichlet, the sequence contains an infinite number of primes.

Here are some other results of mine on Friedman numbers. There are arbitrarily long strings of consecutive Friedman numbers, because of the numbers from 25×102n to 25×102n + 10n – 1 are 10n consecutive Friedman numbers. For example, 250068 = 5002+68. This also shows that there are Friedman numbers ending with any string of digits.

It is also easy to show that if n>60, there is a Friedman number between n and 2n. Zeroes can be added to the right of any of the Friedman numbers 688 = 8 × 86, 1206 = 6 × 201, 1827 = 21 × 87, 3159 = 9 × 351, and 3784 = 8 × 473. The list at the top of the page handles the small cases.

Here are the known small Friedman numbers in other bases:

 Friedman Numbers in Base 2 11001 = 10110 11011 = 110+11 111111 = (11 + 1)11 – 1 1001111 = 11100 – 1 – 1 1010001 = 11100 + 0 + 0 1010011 = 11100 + 10 1100011 = 101010 – 1 1100100 = 101010 + 0 1100101 = 101010 + 1 1111001 = 1 × 101110 1111010 = 101110 + 1 1111011 = 10111 – 1 – 1 1111100 = 10111 – 1 – 0 1111101 = (11 + 1 + 1)10+1 1111110 = (1 + 1)111 – 10 1111111 = (1 + 1)111 – 1 × 1 10001110 = 110010 – 10 10001111 = 110010 – 1 × 1 10011111 = 11 × 110101 10100111 = 110110 – 10 10101001 = 110110 + 0 + 0 10101011 = 110110 + 10 10101111 = 111 × 11001 10111101 = 11 × (10011 – 1) 10111111 = 11 × (1 + 1)110 – 1 11010110 = 11010+1 – 10 11010111 = 11010+1 – 1 × 1 11011000 = 11011 + 0 + 0 + 0 11011001 = 11011 + 0 + 0 + 1 11011010 = 11011 + 0 + 10 11011011 = 11011 + 0 + 11 11011100 = 11011 + 100 11011101 = 11011 + 101 11011110 = 11011 + 110 11011111 = 11011 + 111 11100001 = (10000 – 1)11 11101101 = 11101 – 110 11101110 = 11101 – 101 11101111 = 11101 – 11 – 1 11110001 = 11101 – 10 – 0 11110010 = –1 + 11100+1 + 0 11110011 = 1 + 11100+1 – 1 11110100 = 1 + 11101 + 0 + 0 11110101 = 1 + 11101 + 0 + 1 11110110 = 1 + 11101 + 10 11110111 = 1 + 11101 + 11 11111001 = 10111 × 10 – 1 11111010 = 10111 × (1 + 1) + 0 11111011 = 10111 × (1 + 1) + 1 11111100 = (10111 + 1) × 10 11111101 = 10(1+1)11 – 11 11111110 = (11 + 1)11+1 – 10 11111111 = (11 + 1)11+1 – 1 × 1 100011110 = 1000110 – 11 100011111 = 100011+1 – 1 – 1 100100001 = 1000110 + 0 + 0 100100011 = 1000110 + 10 100101111 = 1100101 × 11 100111001 = (101100 + 1) / 10 100111011 = 1101001 × 11 100111110 = 1101010 × 11 100111111 = 11101 × 1011 101000011 = 1001010 – 1 – 0 101000100 = 1001010 + 0 + 0 101000101 = 1001010 + 0 + 1 101000110 = 1001010 + 10 101000111 = 1001010 + 11 101001111 = 11111 – 1000 101010101 = (1000 – 1)11 – 10 101010110 = (1000 – 1)11 – 1 – 0 101010111 = (101 + 0 + 10)11 × 1 101011001 = (1000 – 1)11 + 10 101011011 = 11111 + 100 + 0 101011110 = 110010 × 111 101011111 = 1110101 × 11 101100111 = 1001110 – 1 – 1 101101001 = 100111+1 + 0 + 0 101101010 = 1001110 + 0 + 1 101101011 = (1011 – 0)10 × 11 101101101 = (11110 + 1 + 0) / 10 101110111 = (1 + 0 + 1 + 1) × 10111 101111010 = (10011 – 1) × 110 101111101 = (10111 – 1) × (10 + 1) 101111110 = 10111 × 11 – 10 101111111 = 10111 × 11 – 1 × 1 110001101 = 1010010 – 11 110001110 = 1010010 – 1 – 1 110001111 = 101001+1 – 1 110010000 = 1010010 + 0 + 0 110010001 = 1010010 + 0 + 1 110010010 = 1010010 + 10 110010011 = 1010010 + 11 110010101 = 11100 × 101 + 0 110101101 = (110010 – 1) × 11 110101110 = (11010+1 – 1) × 10 110101111 = 11010+1 × (1 + 1) – 1 110110011 = (110010 + 1) × 11 110110110 = 1010110 – 11 110110111 = (11 × 111)10 – 10 110111000 = 1010110 – 1 – 0 110111001 = 1010110 × 1 × 1 110111010 = 1010110 + 1 × 1 110111011 = (11 × 111)10 + 10 110111100 = 1010110 + 11 110111111 = (1 + 1)110 × 111 – 1 111001011 = 1111 × 10001 111100010 = 1011010 – 10 111100011 = 1011010 – 1 × 1 111100100 = 101101+1 + 0 + 0 111100101 = 111+100 × 10 – 1 111100110 = (1 + 1 + 1)100+1 × 10 111100111 = 111+100 × (1 + 1) + 1 111101010 = 11110 × 1010 111101100 = (11100 + 1) × 110 111110011 = 101011 / (1 + 1) – 1 111110100 = 101011 / (1 + 1) + 0 111110101 = 101011 / (1 + 1) + 1 111111000 = (10111 + 1) × 100 111111001 = –111 + (1 + 1))1001 111111010 = 1011×11 – 110 111111011 = 11 × (–1 + 1110)1+1 111111100 = (111 + 1)11 – 100 111111101 = (111 + 1)11 – 10 – 1 111111110 = (111 + 1)11 – 1 – 1 – 0 111111111 = (11 – 1)111+1+1 – 1 1000001111 = (11000 – 1)10 – 10 1000110111 = (1000 + 1)10 × 111 1000111010 = 1100010 – 110 1000111011 = 110001+1 – 101 1000111100 = 110001+1 – 100 1000111101 = 110001+1 – 11 – 0 1000111110 = (1000 × 11)1+1 – 10 1000111111 = (1000 × 11)1+1 – 1 × 1 1001001111 = 11 × 11000101 1001010111 = 101010 × 110 – 1 1001011011 = 11 × 11001001 1001011101 = 101110 × 101 + 0 1001011110 = (101010 + 1) × 110 1001100101 = 101100 – 1100 1001101011 = 101100 – 101 – 1 1001101100 = –100 – 1 + 101100 1001101101 = 101100 – 11 – 1 1001101110 = 101100 – 11 – 1 × 0 1001101111 = –1 + 0 + 0 – 1 + 10111+1 1001110000 = 101100 – 1 – 0 – 0 – 0 1001110001 = (100 + 1)11+0+0+0+1 1001110010 = (100 + 1)1×100 + 1 + 0 1001110011 = (100 + 1)1×100 + 1 + 1 1001110100 = 101100 + 11 + 0 + 0 1001110101 = 10111+1 + 100 + 0 1001110110 = (10011 – 1) × 1010 1001110111 = 10111+1 + 110 + 0 1001111001 = 101100 + 1011 1001111010 = 101100 + 1110 1001111011 = (10111 – 1 – 0) × 101 1001111100 = 101100 + 1011 1001111101 = 11110 × 1101 + 0 1001111110 = 10111 × 101 – 10 1001111111 = (100 + 1) × (1 + 1)111 – 1 1010000111 = 110100 / 10 – 1 1010011110 = 1101010 – 110 1010011111 = 110101+1 – 101 1010100011 = 1101010 – 1 – 0 – 0 1010100100 = 1101010 + 0 + 0 + 0 1010100101 = 1101010 + 1 + 0 + 0 1010100110 = 1101010 + 10 + 0 1010100111 = 1101010 + 11 + 0 1010101101 = (1000 – 1)11 × 10 – 1 1010101110 = (101 + 0 + 10)11 × 10 1010101111 = 10 × (10 + 101)11 + 1 1010111011 = 11 × 11101001 1010111100 = 101010 × 111 1010111101 = 11110 – 11100 1010111110 = 1111 × 11010 1010111111 = (1 + 1)110 × 1011 – 1 1011000111 = 11110 – 10010 1011001001 = 11110 – 10000 1011001011 = (110010 – 1) × 101 1011001101 = 11110 – 1100 1011001111 = 11001+1 × 101 – 1 1011010001 = 11110 – 1000 1011010010 = 10 × (–1 + 10100)10 1011010011 = 11110 – 110 – 0 – 0 1011010101 = 11110 – 10 – 10 – 0 1011010110 = (1011 + 0)10 × 110 1011010111 = (10 + 1)10×(10+1) –1 – 1 1011011000 = –1 + 0 + (1 + 10)110 + 0 + 0 1011011001 = –1 + 0 + (1 + 10)110 + 0 + 1 1011011010 = –1 + 0 + (1 + 10)110 + 10 1011011011 = –1 + 0 + (1 + 10)110 + 11 1011011100 = 10 + 1 + (10 + 1)110 + 0 1011011101 = 10 + 1 + (10 + 1)110 + 1 1011011110 = –1 + 110 + 11110 1011011111 = ((10 + 1)101 + 1 + 1) × 11 1011100001 = 11110 + 1000 + 0 1011100011 = 11110 + 1010 + 0 1011100101 = 11110 + 1100 + 0 1011100111 = 11110 + 1110 + 0 1011101001 = 11110 + 10010 1011101010 = 11110 + 10001 1011101101 = (10111 + 0) × 110 – 1 1011101110 = (10111 + 0 × 1) × 110 1011101111 = 10 × 11 × 10111 + 1 1011110001 = 11110 + 11000 1011110011 = 11110 + 11010 1011110100 = 11010 × 10101 1011110101 = 11110 + 11100 1011110111 = 11110 + 11110 1011111001 = 11110 + 10101 1011111010 = (10111 – 1) × 110 + 0 1011111011 = (10111 – 1) × 110 + 1 1011111100 = ((1 + 1)1000 – 1) × 11 – 1 1011111101 = (10(1 + 1)11 – 1) × (10 + 1) 1011111110 = (10111 × 11 – 1) × 10 1011111111 = (1 + 0 + 1)(1 + 1)11 × 11 – 1 1100001110 = 1110010 – 10 + 0 1100001111 = 1110010 – 10 + 1 1100011110 = (101001+1 – 1) × 10 1100011111 = 101001+1 × (1 + 1) – 1 1100101010 = 11100 × 1010 + 0 1100101011 = 11100 × 1010 + 1 1100110011 = (100110 – 1) / 101 1100111111 = (1 + 1)110 × 1101 – 1 1101000111 = 1110110 – 10 – 0 1101001001 = 1110110 + 0 + 0 + 0 1101001011 = 1110110 + 10 + 0 1101001101 = 1110110 + 100 1101001111 = 1110110 + 110 1101011010 = (110010 – 1) × 110 1101011011 = (11011 – 1) × 100 – 1 1101011100 = (11010+1 – 1) × 100 1101011010 = (110010 – 1) × 110 1101011011 = (11011 – 1) × 100 – 1 1101011100 = (11010+1 – 1) × 100 1101011101 = 110011 / 10 – 11 1101011110 = 110011 / 10 – 1 – 1 1101011111 = (110 × 10)11 / (1 + 1) – 1 1101100011 = 100011+1 × 11 + 0 1101100110 = (110010 + 1) × 110 1101101011 = (110 + 1) × (101 + 0)11 1101110001 = 1010110 × 10 – 1 1101110010 = 1010110 × 10 × 1 1101110011 = 1010110 × (1 + 1) + 1 1101110100 = (1010110 + 1) × 10 1101111010 = 11100 × 1011 – 1 1101111011 = (1 + 10)1+11 × 1011 1101111100 = 1 + 1011 × 11100 1101111101 = 1111010 – 111 1101111110 = 10111 × 111 – 10 1101111111 = (1 + 1 + 0)111 × 111 – 1 1110000100 = (100000 – 10)1+1 1110010110 = 1111 × 100010 1110111011 = 1111110 – 110 1110111101 = 111111+1 – 100 1110111110 = 1111110 – 11 – 0 1110111111 = ((1 + 1)101 – 1)1+1 – 1 – 1 1111000000 = (100000 – 1)10 – 1 1111000001 = (10101 – 1)10 + 0 + 0 1111000010 = (100000 – 1)1+1 + 1 1111000011 = (10101 – 1)10 + 10 1111000110 = (1011010 – 1) × 10 1111000111 = (101101+1 × 10) – 1 1111001000 = 1011010 × 10 + 0 1111001001 = 1011010 × 10 + 1 1111001010 = (1011010 + 1) × 10 1111001011 = –1 + 11 × 100101+1 1111001100 = (1 + 1 + 1)100+1 × 100 1111001101 = 11101 × 100 + 0 + 1 1111001110 = 11101 × 100 + 10 1111001111 = 11101 × 100 + 11 1111010011 = 111010 × 101 – 1 1111010100 = 11110 × 10100 1111010101 = 1 + 111010 × 101 1111011000 = (11100 + 1) × 1100 1111011001 = (111010 + 1) × 101 1111011010 = 101011 – 1110 1111100000 = 101011 – 1000 1111100010 = 101011 – 110 – 0 1111100011 = 101011 – 101 × 1 1111100100 = 101011 – 100 × 1 1111100101 = –1111 + 100101 1111100110 = (1011 – 1)11 – 10 – 0 1111100111 = ((1 + 1)11 + 10 + 0)11 – 1 1111101000 = 101011 + 1 × 1 × 0 × 0 1111101001 = (1 + 1 + 1110)10+0+1 1111101010 = 101011 + 1 + 1 + 1 × 0 1111101011 = 1 + 1 + 1 × 1 + 101011 1111101100 = 101011 + 101 – 1 1111101101 = 101011 + 11 + 1 + 1 1111101110 = 101011 + 111 – 1 1111101111 = 111 + (–1 + 1011)11 1111110000 = 110010 × 111 + 0 1111110001 = 110010 × 111 + 1 1111110010 = (1 + 1)1010 – 1110 1111110011 = (1 + 1)1010 – 1101 1111110100 = (1 + 1)1010 – 1100 1111110101 = (11 + 1)101 – 1011 1111110110 = (1 + 1) × 11 × 110110 1111110111 = (11 + 1)101 – 111+1 1111111000 = ((1 + 1)111 – 1) × 1000 1111111001 = –111 + (1 + 1)1+1001 1111111010 = –11 × (1 + 1) + (1 + 1)1010 1111111011 = –11 – 1 + (11 + 1)101 – 1 1111111100 = ((1 + 1)(1+1)11 – 1) × 100 1111111101 = ((1 + 1)11×11 – 1) × 10 – 1 1111111110 = ((1 + 1 × 1)11×11 – 1) × 10 1111111111 = ((1 + 1)111+11 – 1 × 1 × 1

 Friedman Numbers in Base 3 121 = 112 221 = 122 1022 = 202 – 1 1122 = 2 × 211 1211 = 211+1 1212 = 1 + 212 2022 = 220 – 2 2101 = (1 + 1)20 2102 = 220 + 1 10122 = 1012 – 2 10201 = 1012 + 0 11022 = 2 × 2011 11122 = 1 × 121+2 11202 = (1 + 1)20 × 2 11220 = 2 × 2110 12021 = 1102 – 2 12022 = –1 + (20 × 2)2 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12122 = 2 × 2211 12221 = 12 × 212 21021 = 1122 + 0 21102 = 2 × 1012 22021 = 1202 – 2 22100 = 1202 + 0 22101 = 1202 + 1 22102 = 1202 + 2 100111 = (10 + 0 + 1)11 100112 = 2100–1 + 1 101122 = 2 × 1102 – 1 101201 = (200 – 1)1+1 101202 = (200 – 1)2 + 1 101222 = 11 × 2202 102202 = 2002 – 21 102212 = 1 × 12 × 220 102220 = 2002 – 2 – 1 102221 = (201 – 1)2 – 2 102222 = (202 / 2)2 – 1 110022 = 2 × 20011 110122 = 2110 – 2 × 1 110201 = 2110 – 1 × 0 110202 = 2110 + 20 (BH) 110220 = 2 × 20110 111022 = 2012 – 1 – 1 111102 = 2011+1 + 1 111220 = (1 + 2) × 1210 112002 = 2 × 201001 112012 = 10 × 221 – 1 112020 = (1 + 1)20 × 20 112021 = 10 × 221 + 1 112022 = 10 × 221 + 2 112112 = 1121+1 × 2 112200 = 2 × 21100 112202 = 2022 – 1 – 1 112210 = 2021+1 – 1 112211 = (211 – 2)1+1 112212 = (211 – 2)2 + 1 112220 = 2022 + 1 + 1 120221 = 2 × (201+2 – 1) 121002 = 2 × (1020 + 1) 121012 = 2102 – 11 121020 = 2102 – 10 121021 = –1 + 2102 – 1 121022 = 1 + 2102 – 2 121100 = 2101+1 + 0 121101 = 2101+1 + 1 121102 = 2101+1 + 2 121121 = 21 × 111+2 121200 = 10102 / 2 121202 = 2 × (1202 + 1) 121220 = 2 × 22110 122012 = 22 × 2011 122122 = 2112 – 22 122201 = 2112 – 20 122210 = 120 × 212 122211 = 2112 – 2 – 1 122212 = 2 × ((1 + 2)12 + 2) 122220 = (202 + 2)2 – 1 122221 = (2 × (2 × 12 + 1))2 200221 = 2102 – 20 200222 = (2 + 0 + 0) × 222 201121 = 2121+1 + 0 201122 = 2122 + 1 + 0 210022 = 2202 – 1 – 0 211020 = 20 × 1012 211212 = 1211 – 22 211222 = 122×2 – 12 212011 = (21 – 2)0+11 212012 = 122×2 + 0 + 1 212021 = 122×2 + 10 212101 = 1211 + 20 212112 = 12 × (221 – 1) 212122 = 12 × 221 – 2 212201 = 12 × 221 + 0 220221 = 2222 – 10 220222 = (220 + 2)2 – 2 221001 = (1001 – 2)2 221021 = 2 × (2110 – 2) 221101 = 2 × 2110 – 1 221102 = 2 × 211+0+2 221110 = 2 × 2110 + 1 221120 = 2 × (2110 + 2) 221221 = 21 × (2 + 12)2 222200 = 202 × 202 222201 = (1 + 2)20 – 22 222202 = (202 + 2)2 / 2 222221 = (1 + 2)2+2+2 – 2

 Friedman Numbers in Base 4 121 = 112 123 = (1 + 2)3 1203 = 3 × 201 1230 = 3 × 210 1321 = 231+1 1322 = 232 + 1 1331 = (3 + 1 + 1)3 1332 = (3 + 2)3 + 1 2032 = 302 – 2 2213 = 312 – 2 3120 = 123 + 0 3121 = 123 + 1 3122 = 123 + 2 3123 = 3 + 123 3322 = 2 × (3 × 2)3 10132 = 1012 – 3 10201 = 1012 + 0 10221 = 21 × 201 11113 = 131+1+1 11133 = 3 × 1311 11221 = (111 – 2)2 11313 = 113 × 1 × 3 12003 = 3 × 2001 12030 = 3 × 2010 12031 = 1102 – 3 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12232 = 2 × 123 – 2 12233 = –1 + 2 × (2 × 3)3 12300 = 3 × 2100 12303 = 3 × (302 + 1) 12321 = ((1 + 2) × 13)2 12323 = (3 × 13)2 + 2 13023 = 21 × 303 13211 = 2 × 311 + 1 13212 = 2 × 312–1 13222 = 132 × 22 13233 = 313 × 32 13322 = 23×3 – 12 13323 = (3 × 31)2 / 3 13331 = (1 + 1)3×3 – 3 13332 = (–1 + 3)3×3 – 2 21233 = (2 + 3)1+3 – 2 21301 = 113+20 21331 = 31 × 132 21333 = 3 × (123 – 3) 22210 = 1222 + 0 22211 = 1222 + 1 22212 = 1222 + 2 22213 = 1222 + 3 22330 = 32 × 302 22332 = 3 × 3222 23031 = 312 – 30 23102 = 2 × 1032 23112 = 312 – 1 – 2 23113 = –2 + 3(1+1)×3 23120 = 312 – 20 23121 = (23 + 1)2+1 23122 = 32+2+2 + 1 23123 = 2 + 31×2×3 23130 = 312 + 3 + 0 23131 = 312 + 3 + 1 23132 = 312 + 3 + 2 23133 = 312 + 3 + 3 23201 = 312 + 20 23213 = 312 + 32 23322 = 3 × (223 – 2) 31021 = 1312 + 0 31323 = 31+3 × 23 33211 = (21 + 1)3 – 3 33212 = (32 + 1)3 – 2 33220 = ((2 + 3) × 2)3 + 0 33221 = ((2 + 3) × 2)3 + 1 33222 = ((2 + 3) × 2)3 + 2 33223 = 3 + ((3 + 2) × 2)3 33232 = 223 + 3 + 3 33312 = (3 + 3) × 312 33322 = 2 × (23×3 – 3)

 Friedman Numbers in Base 5 121 = 112 224 = 22+4 1232 = 22 × 31 1241 = 241+1 1242 = 1 + 242 1331 = (3 × (1 + 1))3 1332 = (3 × 2)3 + 1 1433 = 143 / 3 1443 = 44 – 13 2123 = 322 – 1 2124 = (24 + 1)2 2244 = (4 × 4 + 2)2 2333 = (–2 + 3 × 3)3 2421 = (41 – 2)2 2423 = 342 + 2 2433 = 4 × 332 3042 = 402 – 3 3421 = 2 × 34+1 3422 = 3 + 422 4243 = 442 – 3 4441 = (4 + 1)4 – 4 10142 = 1012 – 4 10201 = 1012 + 0 10314 = 311 – 40 10413 = 311 + 4 + 0 11424 = 44 × 112 12041 = 1102 – 4 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12104 = 1102 + 4 12144 = 4 × 21 × 41 12320 = 22 × 130 12321 = (2 × 31 – 1)2 12324 = 2 × 3412 12340 = 4 × 310 – 2 12342 = 4 × (1 + 2)2+3 12343 = 4 × 32+3 + 1 12344 = 4 × 31+4 + 2 13041 = 1 × 410 – 3 13043 = 430/3 – 1 13044 = (1 + 3)0+4 × 4 13102 = (1 + 1)20 + 3 13321 = 1132 – 3 13323 = 32 × (1 + 3)3 13324 = (3 × (3 × 4 – 1))2 14111 = 1141+1 14112 = 1142 + 1 14214 = 4 × 2141 14330 = 1 × 30 × 34 20141 = (12 – 1)4 + 0 20311 = 1 × 213 + 0 20312 = 213 + 20 21144 = 4 × 2411 21232 = –2 + 1232 21234 = (1 × 2 × 34)2 21243 = 1232 + 4 21311 = 2 × (–1 + 311) 21313 = 2 × (13 + 1)3 21314 = 2 × 143 + 1 21414 = 24 × 411 22041 = (2 × 40 – 1)2 22314 = 21 × (3 × 4)2 22400 = (2 × 40)2 + 0 22401 = (2 × 40)2 + 1 22402 = 2 + (2 × 40)2 22403 = (2 × 40)2 + 3 22404 = (2 × 40)2 + 4 23211 = (132 – 1)2 23212 = 2132 / 2 23334 = (3 × 4)3 – 32 23341 = (3 × 4)3 – 12 23343 = –2 – 3 (3 × 4)3 23344 = (23 + 4)3 – 4 23402 = 223 – 40 23403 = (23 + 4 + 0)3 23412 = 223 + 1 × 4 23434 = (3 × 4)3 + 24 24024 = (40 + 42)2 24132 = 31 × 422 24244 = 424 × 24 24330 = 4 × 3320 24343 = 42 × (34 + 3) 24344 = (34 + 44)2 30234 = (40 – 2)3 / 3 30444 = (403 – 4) / 4 31042 = 1402 – 3 31134 = 4 × (133 – 1) 31142 = 23×4–1 – 1 31143 = (3 – 1)–1+4×3 31144 = 13 × 44 + 1 31204 = 2 × (410 + 3) 31422 = 221 + 43 32221 = 312 – 2 / 2 32222 = 3(22+2)/2 32224 = 324/2 + 2 32233 = 3 × (32×3 + 2) 32242 = (24 – 2 / 2)3 32314 = (134 + 3)2 33204 = (4 × (30 – 3))2 33222 = 23 × 322 34021 = 124 – 30 34041 = (3 + 4)4 – 10 34042 = (20 – 3)4 – 4 34043 = (3 + 4)0+4 – 3 34101 = (3 + 4)10–1 34102 = 124 + 30 34210 = 20 × 31+4 34212 = 34 × 212 34412 = 124 + 34 34423 = 4 × ((2 + 3)4 – 3) 40214 = 20 × 44 – 1 40332 = 2 × (3 + 3)4 + 0 41122 = 2 × 214–1 43124 = 4 × 143 – 2 43131 = 4 × 33×(1+1) 43132 = 4 × 33×2 + 1 43134 = 4 × 143 + 3 43234 = 43 × 23+4 44231 = 3 × (41+4 – 2) 44233 = –4 + 42+3 × 3 44242 = 44 × (2 + 4) × 2 44301 = 3 × 410 + 4 44313 = 413 / 3 – 4 44314 = 3 × 4 × (44 + 1) 44331 = 413 / 3 + 4

 Friedman Numbers in Base 6 24 = 24 52 = 25 121 = 112 124 = 4 × 21 133 = 3 × 31 143 = –1 + 43 144 = 44–1 213 = 132 1043 = 34+1 + 0 1053 = 35 + 10 1135 = 5 × 131 1204 = 4 × 201 1224 = 2 × 412 1240 = 4 × 210 1242 = 2 × 421 1252 = 5 × 122 1303 = 3 × 301 1330 = 3 × 310 1352 = 312 – 5 1353 = 3 × 315 1423 = 23 × 41 1425 = 21 × 45 1524 = 4 × 251 1533 = 3 × 351 2212 = 2(2+1)2 2213 = 232 + 1 2235 = 352 – 2 2241 = (41 – 2)2 2355 = 35 × 52 2400 = 402 + 0 2401 = 402 + 1 2402 = 2 + 402 2403 = 402 + 3 2404 = 402 + 4 2405 = 402 + 5 2514 = 54 – 2 – 1 2515 = –2 + 55–1 2521 = 1 × 52+2 2534 = 54 + 32 2535 = (2 + 53) × 5 2544 = 5 × 44 / 2 3213 = 32+1+3 3214 = 34+2 + 1 3215 = 35+1 + 2 3453 = 5 × 433 3513 = 51 × 33 3524 = 33 × 51 4052 = 502 – 4 4352 = 45 – 32 4415 = 44+1 – 5 4424 = 44+2 / 4 4431 = 44+1 + 3 4432 = 4 + 43+2 4435 = 45 + 4 + 3 4452 = 45 + 24 4504 = 45 + 40 4515 = 45 + 51 5204 = 542 + 0 5325 = 5 × (35 – 2) 5343 = 5 × 34 × 3 5432 = (54 – 3) × 2 5442 = 54 × 4 / 2 5532 = 5 × 25+3 10125 = 1012 – 5 10201 = (100 + 1)2 10231 = 201 × 31 10321 = 301 × 21 10414 = 401 × 14 10452 = 504 × 12 11013 = 1031+1 11224 = (112 – 4)2 11225 = (12 × 5)2 + 1 11253 = 121 × 53 11305 = 1301 × 5 11343 = 141 × 33 11350 = 1310 × 5 11513 = 1 × 11 × 35 11535 = 5 × (1 + 1 + 5)3 11543 = 5 × 113 + 4 12004 = 2001 × 4 12024 = 4012 × 2 12040 = 2010 × 4 12042 = 4021 × 2 12051 = 1102 – 5 12100 = 1102 + 0 12101 = 1102 + 1 12102 = 1102 + 2 12103 = 1102 + 3 12104 = 1102 + 4 12105 = 1102 + 5 12144 = (12 – 1) × 44 12204 = 4102 × 2 12240 = 4120 × 2 12254 = 4125 × 2 12321 = (113 – 2)2 12342 = 242 × 31 12353 = 3 × (53+1 – 2) 12354 = 3 × (54 – 1) – 2 12400 = 400 × 21 12402 = 4201 × 2 12403 = 203 × 41 12420 = 4210 × 2 12520 = 50 × 122 12521 = 2 × 512 – 1 12522 = (–1 + 25)2 × 2 12532 = 12 × (35 – 2) 12542 = 4251 × 2 12544 = 4 × 4 × 152 12554 = (1 + 5)5 / 4 – 2 13003 = 3001 × 3 13030 = 3010 × 3 13053 = 3015 × 3 13132 = (13 + 1)3 × 2 13213 = 3 × 231+1 13220 = 2102 / 3 13235 = 215 – 3 × 3 13242 = 214 – 2 × 3 13244 = 123 × 4 – 4 13245 = 1 × 2 × 45 – 3 13251 = 2 × (1 + 3)5 – 1 13252 = (–1 + 3)2×5 × 2 13253 = 23+3+5 + 1 13254 = 2 × 45 + 3 – 1 13255 = 25+5+1 + 3 13300 = 3100 × 3 13325 = 215 + 33 13435 = 53 × 43 – 1 13452 = 352 + 1 × 4 13530 = 3150 × 3 13543 = 13 × (35 – 4) 13553 = 3155 × 3 14023 = 401 × 23 14043 = (–1 + 4 + 0)4+3 14121 = (111 + 4)2 14124 = 221 × 44 14125 = 1152 + 4 14230 = 230 × 41 14250 = 210 × 45 14252 = 5124 × 2 14322 = 431 × 22 14343 = 3 × 3 × (44 – 1) 14344 = 44 × (34 + 1) 14352 = 35 × 142 14413 = 13 × (44 + 1) 14522 = 5241 × 2 14523 = 4215 × 3 15024 = 504 × 21 15041 = (1 + 5 + 0 + 1)4 15042 = (5 + 0 + 2)4 + 1 15134 = 114 + 53 15204 = 2501 × 4 15213 = 513 × 21 15224 = 5412 × 2 15234 = 5 × 31 × 42 15240 = 2510 × 4 15242 = 5421 × 2 15243 = 2143 × 5 15303 = 3501 × 3 15323 = 32 × (53 – 1) 15324 = (–1 + 5) × (3 + 2)4 15330 = 3510 × 3 15343 = 3 × 43 × 51 15344 = 1 × 4 × (54 + 3) 15353 = 3515 × 3 15404 = 4 × (54 + 10) 15432 = 5 × (123 – 4) 15524 = 2551 × 4 15533 = 3551 × 3 15542 = 2 × ((1 + 5)4 – 5) 20412 = 224 – 1 – 0 21300 = 1302 + 0 21301 = 1302 + 1 21302 = 2 + 1302 21303 = 1302 + 3 21304 = 1302 + 4 21305 = 1302 + 5 21324 = 3221 × 4 21354 = 314 × 52 21435 = 2431 × 5 21455 = 251 × 45 22120 = 20 × 212 22245 = 2 × (2 – 2) + 45 22251 = 52+2+1 + 2 22253 = 53+2 + 2 + 2 22255 = 2 + 2 + 2 + 55 22304 = (304 / 2)2 22305 = 52+3 + 20 22355 = 22 × 3 + 55 22452 = 2 × (2 × 4 × 5)2 22505 = 55 + 220 23013 = (31 × 3)2 + 0 23254 = 22× (35 – 4) 23324 = (2 + 33)2 × 4 23325 = 233 – 2 × 5 23333 = 233 – 3 – 3 23341 = (3 × (4 + 1))3 – 2 23343 = (2 + 3 × 3 + 4)3 23344 = 233 + 4 / 4 23345 = (3 × 5)3 + 4 – 2 23350 = (50 / 2)3 + 3 23351 = 233 + 5 – 1 23353 = 2 × 3 + (3 × 5)3 23424 = 442 × 32 23514 = 14 × (2 + 5)3 23550 = 350 × 52 24135 = (423 – 1) / 5 24332 = (3 + 4) × 232 24353 = (3 × 4 × 5)2 – 3 24513 = 5 × (2 × 4 + 1)3 24533 = 423 × 35 25044 = 404 × 52 25045 = 55 + 402 25121 = (2 × 51 – 1)2 25210 = (10 × 5)2 + 2 25221 = 21 × 252 25350 = 50 × (53 + 2) 25413 = 413 × 52 25434 = 2 × 5 × 44 – 2 25440 = (50 / 2) × 44 25444 = (54 + 44)2 25551 = (5 + 1)5 / 2 – 5 30541 = 45+1 – 3 – 0 30542 = (3 + 0 + 5)4 – 2 30543 = (3 + 5)4 – 30 30544 = ((–3 + 0 + 5) × 4)4 30545 = (3 + 5)4 + 50 31155 = 3511 × 5 31321 = ((3 + 1)3 + 1)2 31324 = (43 + 1)2 + 3 32114 = 2 × (311 – 4) 32125 = 2 × 32+5 – 1 32130 = 3 × 2 × 310 32214 = 2 × (213 + 4) 32242 = 2342 / 2 32441 = (43 + 4 – 1)2 33213 = 31 × 32+3 33215 = 31 × 35 + 2 33224 = (34 × 3 + 2)2 33233 = (333 – 3) / 2 34112 = 2 × (114 – 3) 34120 = 3 × 1042 34122 = (3 × 2 + 1)4 + 2 34124 = 2 × ((3 + 4)4 + 1) 34421 = (42 + 1)3 – 4 34425 = (4 × 2 + 4 + 5)3 34433 = (33 – 4)3 + 4 34530 = 4330 × 5 35052 = (5 × 2)3 × 5 + 0 35130 = 510 × 33 35233 = 33 × (35 – 2) 35255 = 535 × 52 35343 = 35 × (3 + 4) × 3 35405 = -3 + 5 × 40+5 35412 = 1 × (3 + 2) × 45 35415 = 3 + 5 × 41×5 35421 = (3 + 2) × (45 + 1) 35425 = 5 × 45 + 32 35513 = 515 × 33 41345 = 134 – 45 41354 = 431 × 54 41453 = 441 × 53 41525 = 521 × 45 42013 = (41 × 3)2 + 0 42420 = 2042 – 4 43234 = (34 – 4)2 – 3 43241 = 1 × (34 – 4)2 43521 = 41 × (35 – 2) 44240 = 40 × (4 × 4)2 44521 = (4 × 4 × 5 – 1)2 44531 = 14 × (54 – 3) 45515 = 5 × ((5 + 1)4 – 5) 45532 = 534 × 52 45551 = 5 × (5 + 1)4 – 5 50213 = 35+0+2+1 50233 = 35+3 + 20 51344 = 44 × (35 + 1) 51423 = 315–2 – 4 51430 = (40 – 5)3 – 1 51431 = (5 × 1 × 4 – 1)3 51455 = (5 + 1 + 5) × 54 51514 = 15 × (54 + 1) 52343 = 45 × 32+3 52345 = 45 × 35 + 2 53141 = 11 × (45 + 3) 53241 = (34 + 5 – 1)2 53250 = 50 × (35 – 2) 53430 = 5 × 34 × 30 54123 = (34 + 5)2 – 1 54124 = ((4 – 1)4 + 5)2 54212 = 45 × 212 54320 = 20 × (54 – 3) 54413 = (54 × 4 – 1) × 3 54420 = 40 × 54 / 2 54432 = 4 × (54 × 3 + 2) 54433 = (54 × 4 + 3) × 3 55320 = 50 × 25+3 55515 = -5 × 5 + (5 + 1)5 55542 = (4 + 2)5 – 5 – 5 55551 = (5 + 15)5 – 5 55553 = (5 / 5 + 5)5 – 3

 Friedman Numbers in Base 7 121 = 112 143 = 34 – 1 144 = (–1 + 4)4 264 = 4 × 62 514 = (5 – 1)4 1155 = 15 × 51 1253 = 312 – 5 1263 = 2 × 36–1 1331 = (1 + 1)3×3 1332 = 23×3 + 1 1452 = 51 × 42 1541 = 54 – 11 1544 = –1 + 54 – 4 1545 = 1 + 54 – 5 1551 = –1 + 55–1 1552 = 1 × (5 × 5)2 1553 = 1 + 5 × 53 1614 = 64 / (1+1) 2061 = (2 + 1)6 + 0 2063 = 36 + 2 + 0 2314 = 412 + 3 2343 = 4 × (2 × 3)3 2422 = –2 + 422 2424 = 424–2 2534 = 432 + 5 2542 = (45 – 2)2 2635 = (5 × 2)3 + 6 2640 = 40 × 62 2654 = 26+4 – 5 2662 = 26+6–2 3354 = 5 × 3 × 34 3425 = 2 × (54 – 3) 3464 = –34 + 64 3531 = (3 + 3)5–1 4323 = 3 × (4 × 2)3 4625 = (64 – 5)2 5062 = 602 – 5 5446 = (65 + 4) / 4 5622 = (65 – 2)2 6243 = (6 / 2)4+3 6265 = 652 – 6

 Friedman Numbers in Base 8 33 = 33 121 = 112 125 = 5 × 21 143 = 3 × 41 251 = 152 257 = 7 × 52 326 = 63 – 2 363 = 36 / 3 527 = 75–2 1133 = 3 × 311 1205 = 5 × 201 1250 = 5 × 210 1326 = 132 × 6 1327 = 37–1 – 2 1331 = 33×(1+1) 1332 = 1 + 33×2 1356 = 5 × 6 × 31 1403 = 3 × 401 1430 = 30 × 41 1626 = 21 × 66 1724 = 7 × 214 2147 = 7 × 241 2204 = 422 + 0 2342 = (2 + 3)4 × 2 2345 = 2 × 54 + 3 2346 = 3 × 642 2372 = 32 × 72 2416 = 64 – 1 × 2 2534 = (2 + 5)3 × 4 2544 = 5 × 424 2570 = 70 × 52 2642 = 462 – 2 2644 = 464–2 2662 = (2 / 6)–6 × 2 2754 = 472 – 5 3245 = 35 × 34 3275 = 5 × (73 + 2) 3534 = 3 × (54 + 3) 4213 = 34+2+1 4217 = 4 + (2 + 1)7 4237 = 37 + 24 4334 = 34 × 34 4527 = 74 – 5 × 2 4534 = (3 + 4)4 – 5 4537 = 74 – 5 + 3 4541 = (14 – 5)4 4572 = 74 + 52 4576 = 5 × 746 4654 = 4 × (–6 + 54) 5374 = 54 + 37 5676 = 6 × 765 5726 = 672 + 5 6065 = (–60 + 6)5 6072 = 702 – 6 7246 = (7 – 2)4 × 6

 Friedman Numbers in Base 9 121 = 112 134 = 4 × 31 314 = (3 + 1)4 628 = 8 × 26 1304 = 4 × 301 1326 = 2 × 613 1340 = 31 × 40 1354 = 45 – 1 × 3 1357 = 1 × (7 – 3)5 1362 = 2 × 631 1363 = 3 × (1 + 6)3 1438 = 18 × 43 1456 = 4 × 6 × 51 2086 = 20+8 × 6 2132 = (32 – 1)2 2136 = 6 × 321 2467 = 472 – 6 2472 = –2 + 472 2474 = 474–2 2725 = 27+5 / 2 2737 = –27 + 37 3254 = –3 + (2 + 5)4 3247 = 74 – 32 3257 = 75+2–3 3454 = 5 × (4 + 4)3 3458 = 5 × 83 + 4 3672 = (2 × 7)3 – 6 3678 = (7 × (8 – 6))3 4126 = 612 – 4 4252 = (4 + 2 / 2)5 4357 = 37 + 45 5485 = 84 – 55 6280 = 80 × 26 6827 = 782 + 6 7082 = 802 – 7 8836 = 38 – 6 × 8 8873 = 38 – 7 – 8

 Friedman Numbers in Base 11 121 = 112 2A9 = (A + 9)2 603 = 36 + 0 1163 = 361+1 1533 = 53 × 31 15A2 = 2A+15 16A3 = A × (63 + 1) 1974 = 74 + 91 1A18 = A × (1 + 1)8 2343 = 3 × 42+3 2345 = 2+ 3 × 45 2494 = (44 + 9)2 2783 = 82 × 37 2794 = 29 + 4 × 7 2883 = (8 × 8 – 3)2 3094 = (– 30 + 9)4 3317 = (3 – 1) × 37 3518 = 18 × 35 362A = 632 – A 3642 = 632 + 4 3879 = 7 × (93 – 8) 48A2 = 4 + (8 × A)2 4A18 = (A – 1)4 – 8 4A34 = 34+4 + A 4A38 = 4 + A + 38 4A98 = 94 + 10 × 8 5287 = 725 × 8 5327 = 27 × 35 5478 = 7 × (45 + 8) 7739 = (7 + 7) × 93 8A29 = 9A2 + 8 90A2 = A02 – 9 9A25 = A52 – 9 A428 = 4A × 28

 Friedman Numbers in Base 12 121 = 112 127 = 21 × 7 135 = 31 × 5 144 = 41 × 4 163 = 61 × 3 346 = 34 × 6 368 = 86–3 376 = 73 × 6 441 = (4 + 1)4 445 = 54 + 4 114A = 141 × A 1169 = 161 × 9 1207 = 201 × 7 1228 = 21+2+8 1229 = 1 + 22+9 122A = 2 × (2A + 1) 122B = 1 + 2 + 2B 123B = 2B + 13 1270 = 210 × 7 12B9 = 2B + 91 1305 = 301 × 5 1323 = 32×3+1 1350 = 310 × 5 1404 = 401 × 4 1428 = 814 × 2 1440 = 410 × 4 1476 = 74 – 6 – 1 1477 = 74 – 7 + 1 1481 = (8 – 1)4 × 1 1482 = 841 × 2 1544 = 1 × 54 × 4 1603 = 601 × 3 1630 = 610 × 3 1826 = 81 × 26 1924 = 91 × 24 1A28 = A8 × 21 1B53 = (-1 + B + 5)3 2379 = 927 × 3 2448 = (2 × 4)4 – 8 2452 = 542 – 2 2454 = -2 + 45 × 4 2468 = 46 + 2 × 8 2525 = 252 × 5 2541 = (54 + 1)2 2545 = 552 + 4 257A = 5 × 7 × A2 2636 = 6 × (36 – 2) 2779 = 79 × 72 2815 = 582 + 1 2942 = (-2 + 9)4 × 2 2A84 = (A4 – 8) / 2 2B36 = 326 × B 3166 = 631 × 6 3266 = 63 × 62 3460 = (60 / 4) 3548 = 834 × 5 35A6 = 6 × (A3 + 5) 3760 = 730 × 6 37B6 = 73B × 6 3963 = 39 / 3 – 6 3B76 = 7B3 × 6 416B = 461 × B 45A8 = 548 × A 46A8 = 6A4 × 8 47BA = 4A × B7 4892 = 2 × (84 – 9) 48A8 = (A – 2) × 84 4969 = 649 × 9 5513 = 55 × 1 × 3 5788 = 857 × 8 57BA = 75 × B × A 5954 = (9 + 5 / 5)4 597B = 9B5 × 7 5B14 = (11 – 1) × 45 5B22 = 2B + 2 × 5 6946 = 9 × (64 + 6) 7651 = (57 + 1) / 6 7A28 = 8A × 27 95A2 = 29+5 + A 9B2A = AB2 + 9 A0B2 = B02 – A A454 = 5A × 44 BA24 = A × 2B – 4

 Friedman Numbers in Base 13 121 = 112 237 = 27 × 3 24A = A2 × 4 1245 = (1 + 5)4 × 2 1246 = 64 × 2 + 1 1353 = 33 × 51 1524 = (1 + 2) × 45 1559 = 1 × 55 – 9 1565 = 1 × 56 / 5 1623 = 12 × 63 173A = 73 × A – 1 173B = 73 × (B – 1) 1829 = (8 – 1) × 29 1B27 = 2B+1 – 7 1B31 = (3 – 1)B+1 1B3C = (3 – 1)C + B 22B8 = 28 × B2 2370 = 30 × 27 248A = (4 × (8 + A))2 24A0 = 40 × A2 256B = B5 × 62 2653 = (56 + 3)2 2659 = 592 + 6 2866 = 66 / 8 – 2 2868 = 68–2 / 8 2A3B = 2B × 3 – A 2CA9 = (C – 9)A–2 31B9 = 39 × B1 3845 = (5 × (8 – 4))3 3C24 = (34 + C)2 3C25 = (C3 + 2) × 5 3C5B = C3 + 5 × B 3C96 = 93 × C – 6 3CA9 = 93 × C + A 426C = ((C + 4) × 6)2 4719 = (9 + 1)4 – 7 471A = (A + 7 – 1)4 5682 = 862 + 5 7155 = 55+1 – 7 7156 = 57–1 – 6 715C = (C – 7)5+1 7165 = 56 + 7 – 1 7166 = 7 + (–1 + 6)6 75C4 = (7 – 5)C × 4 77C5 = 75 – 7 × C A3B7 = (A + 7) × B3 AC2B = BC2 + A B0C2 = C02 – B

 Friedman Numbers in Base 14 121 = 112 128 = 21 × 8 135 = 1 × 35 144 = 1 × 44 173 = 71 × 3 237 = (3 × 7)2 248 = 82 × 4 288 = 82 × 8 294 = 42 × 9 396 = 36 – 9 3A1 = (A – 1)3 C37 = 7C/3 1196 = (1 + 1)9 × 6 11B6 = 16 × B6 1208 = 201 × 8 126A = 2A × 61 1280 = 21 × 80 1324 = (43 – 1)2 1350 = 10 × 35 1356 = 36 × 51 1375 = 371 × 5 137A = 73 × A + 1 1440 = 10 × 44 1447 = 41 × 47 1452 = (45 – 1)2 1488 = 41 × 8 × 8 14AC = 1C4 × A 16C2 = 1 × 2C – 6 16C8 = 1 × (8 – 6)C 1703 = 701 × 3 1730 = 71 × 30 1832 = (18 × 3)2 1B84 = 41 × B × 8 1C29 = C9 × 21 1C83 = 8C1 × 3 218C = 182 × C 21A8 = (A + 8)2+1 234C = (2 × 4)3 × C 2408 = 802 × 4 2480 = 40 × 82 2569 = (–2 + 5)6 × 9 2880 = 82 × 80 2940 = 90 × 42 29A4 = (9 × A – 4)2 2A3A = AA2 / 3 2B37 = B3 × 72 2D65 = D × 56–2 2D93 = 2D – 9 × 3 2DA3 = 2D – A + 3 2DAD = 2D + A – D 2DB1 = 2D – 1B 2DC0 = 2D + C + 0 2DC1 = 2D + C + 1 2DC2 = 2D + C + 2 2DC3 = 2D + C + 3 2DC4 = 2D + C + 4 2DC5 = 2D + C + 5 2DC6 = 2D + C + 6 2DC7 = 2D + C + 7 2DC8 = 2D + C + 8 2DC9 = 2D + C + 9 2DCA = 2D + C + A 2DCB = 2D + C + B 2DCC = 2D + C + C 2DCD = 2D + C + D 2DD1 = 2D + 1D 328C = 38–2 × C 378C = (C + 7) × 83 384C = C83 × 4 38D2 = 3 × D × 28 3904 = (30 + 9)4 415B = (45 – 1) × B 458C = C48 × 5 4BCA = BC/4 × A 517D = 571 × D 5962 = (65 + 9) × 2 59B7 = B59 × 7 5B97 = B95 × 7 61A7 = 76–1A 7645 = (46 – 7) × 5 8A2C = A28 × C 91AC = A91 × C 997B = B99–7 A929 = ((A + 9) × 9)2 BD2C = CD2 + B C0D2 = D02 – C D32C = 3D2 × C

 Friedman Numbers in Base 15 26 = 62 121 = 112 136 = 31 × 6 154 = 51 × 4 336 = –3 + 36 339 = 39–3 484 = 84 / 4 1144 = 411 × 4 11AA = 1A1 × A 1214 = 421+1 1235 = 21 × 53 12B3 = (1 + 2) × B3 12DA = 21 × A × D 12DC = 1D × C2 1306 = 301 × 6 132C = 2C + 3 – 1 1360 = 31 × 60 1469 = 94–1 × 6 1504 = 501 × 4 152A = A15 × 2 1540 = 51 × 40 15A2 = A51 × 2 163E = 6E × 31 17A3 = (A – 1)3 × 7 192A = 91 × 2A 1DB3 = D1 × B × 3 1E26 = (1 + 2)E–6 2428 = 842 / 2 2486 = (8 – 2)4 × 6 24A8 = 8A2 × 4 252E = 5E2 – 2 2585 = (5 × 8)2 × 5 258D = 52 × 8 × D 2600 = 602 + 0 2601 = 602 + 1 2602 = 2 + 602 2603 = 602 + 3 2604 = 602 + 4 2605 = 602 + 5 2606 = 602 + 6 2607 = 602 + 7 2608 = 602 + 8 2609 = 602 + 9 260A = 602 + A 260B = 602 + B 260C = 602 + C 260D = 602 + D 260E = 602 + E 2662 = (–2 + 6)6 × 2 2757 = (75 + 7) / 2 28C6 = 8C2 × 6 2B87 = B2 × 8 × 7 2E6A = AE–6 / 2 324D = (2 × D – 4)3 33D5 = (D3 + 3) × 5 345C = C53 × 4 34BA = BA × 43 36C9 = 6C/3 × 9 3839 = (3 + 8)3 × 9 3CAD = A3 × D – C 3D4A = DA × 43 3D66 = (D3 – 6) × 6 3D96 = D3 × 6 + 9 3DE6 = D3 + E × 6 3EBA = (B3 + E) × A 450B = B4 – 5 – 0 4511 = (11 – 5)4 452B = B4 + 52 4593 = (95 + 3) / 4 4639 = 49 × 63 491C = 9 × C4–1 4961 = (4 + 1)6 – 9 4965 = (9 – 4)6 – 5 496A = (A + 4 – 9)6 496E = 4 + (–9 + E)6 4CD2 = 2C × 4 + D 4EA7 = 74A/E 58DA = 85D × A 6224 = (6 × 2)4 – 2 6226 = (6 × 2)6–2 622C = C2+2 + 6 624C = C4 + 62 7DCB = C7 × D × B 86AC = A86 × C 8B74 = B78/4 8DA5 = D58 × A AC21 = (C1 + A)2 AC1B = CB2 + A B23C = 3B2 × C BDEC = EDB × C C20A = 2C + 0 × 10 C42E = (C4 – E) × 2 C452 = C4 × 2 + 5 CE2D = DE2 + C D0E2 = E02 – D DC56 = (D – C + 5)6 E326 = (E3 + 6)2 E329 = E92 + 3

 Friedman Numbers in Base 16 121 = 112 129 = 21 × 9 145 = 41 × 5 183 = 81 × 3 27D = 72 × D 57C = 75 × C 1209 = 201 × 9 1236 = 612 × 3 1263 = 621 × 3 1290 = 21 × 90 1405 = 401 × 5 1450 = 41 × 50 15F9 = F5–1 / 9 1803 = 801 × 3 1830 = 81 × 30 192C = 219 × C 19DE = 1D9 × E 1E2A = 21 × EA 2373 = 372 × 3 27D0 = D0 × 72 2BA8 = (2A – 8) × B 2F78 = F8 × 72 32FA = 3A × F2 3342 = (3 × 3)4 × 2 3A27 = 7A2 + 3 3F48 = 438 × F 41A7 = 74+1A 41EB = 4B × F1 4628 = 862 + 4 467E = (64 – 7) × E 46E0 = 64 × E + 0 46E1 = 64 × E + 1 46E2 = 64 × E + 2 46E3 = 64 × E + 3 46E4 = 64 × E + 4 46E5 = 64 × E + 5 46E6 = 64 × E + 6 46E7 = 64 × E + 7 46E8 = 64 × E + 8 46E9 = 64 × E + 9 46EA = 64 × E + A 46EB = 64 × E + B 46EC = 64 × E + C 46ED = 64 × E + D 46EE = 64 × E + E 46EF = 64 × E + F 4D39 = –4 + D3 × 9 4D3D = D3 × (D – 4) 52B7 = (5 × B)2 × 7 55C7 = 55 + C × 7 57C0 = 75 × C0 59DC = D5 × C × 9 5EC6 = 6C5 × E 5F7C = 7F5 × C 618F = 681 × F 672B = 76–2 × B 76BA = B6 × A7 7DFA = D7 × A × F 85FC = 8C × F5 DF2E = EF2 + D E0F2 = F02 – E E695 = 95 – E – 6

Trevor Green proved that there are infinitely many Friedman numbers in every base by considering numbers of the form 1000...02000...01=1000...012+0+0+...+0 in bases larger than 2 and numbers of the form 1000...01000...0001=1000110+0+0+...+0 in base 2.

Trevor Green also writes: "25 is a Friedman number in bases 2, 3 and 4 as well as base 10. What other numbers are Friedman numbers in more than one base, or in an unusually large number of bases? What numbers are not Friedman numbers in any base?"

Trevor Green has found several other strings which are Friedman numbers in all large bases, such as 102030201 = (10301 – 200)2 and 1367631 = (117 – 6)(6+3)/3. Also, he found the series 121 = 112, 12321 = (113 – 2)2, 1234321 = (1143 – 32)2, 123454321 = (11543 – 432)2, . . . .

Trevor Green noticed that when a>1 and b>2, 1ab is a Friedman number in base a(b–1), since it can be written 1ab = b×a1. He also noted that when a>1 and b>1, if we let c=a×b and d=a2×b, then acd is a Friedman number in base b×(d–1), since it can be written acd = c×da.

In 2007, Trevor Green sent me a list of the 3–digit Friedman numbers in all bases up to 100 that are not part of a known family. They can be found here. A month later he sent me some of the patterns from the list.

Robert Happelberg invented the concept of Roman Friedman Numbers. Here are the ones we found less than 100:

 Roman Friedman Numbers VIII = IV × II XVIII = IV × II + X XXVII = IX × (X/V + I) (BH) XXVIII = IV × II + XX XXXIII = XI × (X/X + II) XXXVI = VIXX/X XXXVII = IX × (X/V + I) + X (BH) XXXVIII = IV × II + XXX XLIV = L – V – IX XLVI = L – V + IX XLVII = L – X/V – I × I (BH) XLVIII = IV × II + XL XLIX = L – IXX LVIII = IV × II + L LXVIII = IV × II + LX LXXIV = L × XV / X – I (BH) LXXV = L / X × XV (BH) LXXVI = L / X × XV + I (BH) LXXVII = L / X × XV + II (BH) LXXVIII = L / X × XV + III (BH) LXXXI = IXX×X/L LXXXII = IXX×X/L + I LXXXIII = IXX×X/L + II LXXXIV = LX / X × XIV (AF) LXXXVI = L × XV / X + XI LXXXVII = L × XV / X + XII LXXXVIII = IV × II + LXXX LXXXIX = X × (X – IL) – X/X XCIV = C – V – IX XCVI = C – V + IX XCVII = C – X/V + I×I XCVIII = IV × II + XC XCIX = C – IXX

In 2013, Alan Frank found the first non-trivial nice Roman Friedman number. Then Bryce Herdt found several more.

If you can extend any of these results, please e–mail me. Click here to go back to Math Magic. Last updated 6/29/13.