Problem of the Month (April 2012)

Friedman numbers, numbers that can be written in a non-trivial way using their own digits using only +, –, ×, /, ^, concatenation of digits, and any number of parentheses, was the August 2000 Math Magic problem. There are several other web sites allowing differing operations: Vampire Numbers, Narcissistic Numbers, Wild Narcissistic Numbers, Pretty Wild Narcissistic Numbers. This month we investigate some related questions with what we will generally call self-reproducing numbers.

#1. Which numbers can be written using only one or two operations? We allow !, √, and decimal points as well, since they are popular choices among recreational math enthusiasts. What about Fibonacci numbers, Catalan numbers, or binomial coefficients? What other operations yield self-reproducing numbers?

#2. Which numbers n can be written using only the digits from n-1 or n+1? We call these almost Friedman numbers.

ANSWERS

Zsbán Ambrus, Mark Mammel, Joe DeVincentis, and Bryce Herdt sent solutions.

#1.

As before, numbers are nicer if they can written with the digits in the original order, and below these are shown in red.

Exponentiation
25 = 52 4096 = (46)90 (MM) 15625 = (56)125 19683 = (39)168 59049 = (95)409
78125 = (57)182 117649 = ((76)14)19 390625 = (523)609 (MM) 1953125 = ((59)13)125 5764801 = ((78)156)40
10077696 = (((69)16)70)70 40353607 = (((73)3)40)506 43046721 = (((34)4)60)127 134217728 = ((832)147)127 (JD) 244140625 = (((52)6)144)204
387420489 = ((32)9)807448

Exponentiation and Addition
125 = 51+2 216 = 62+1 289 = (8 + 9)2 343 = (3 + 4)3 347 = 73 + 4
736 = 7 + 36 2187 = (2 + 18)7 3125 = (31 + 2)5 3375 = (3 + 5 + 7)3 3378 = (7 + 8)3 + 3
13824 = (81 + 24)3 14641 = (1 + 4 + 6)41 14645 = (5 + 6)4 + 41 14647 = (4 + 7)4 + 61 15626 = 56 + 126
15627 = (56)17 + 2 15628 = 56 + 18 + 2 15631 = (51+1)3 + 6 15632 = (52)3 + 6 + 1 15641 = 56 + 41+1
15642 = 1 + 56 + 42 15661 = 56 + 61+1 15662 = 1 + 56 + 62 16384 = 46+138 16815 = (1 + 6)5 + 81
17536 = 1 + 75 + 36 19732 = 39 + 712 32771 = 3 + 27+7+1 32775 = 23+5+7 + 7 32778 = 27+8 + 7 + 3
38416 = (613 + 8)4 44977 = (7 + 7)4 + 94 46660 = (4 + 66)60 46661 = 66 + 4 + 16 59050 = 95 + 50 + 0
59051 = 95 + 50 + 1 59052 = 95 + 50 + 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 78225 = 57 + (2 + 8)2
104976 = (1 + 9 + 7 + 60)4 (JD) 117650 = 11 + (76)50 1679616 = 6(7+1)1966 (JD) 2097152 = (27)2+1+09+5 (JD) 2097153 = 20+7+9+5 + 13 (JD)
16777216 = (16+7+7 + 7)21+6 (JD)

Exponentiation and Subtraction
127 = 27 – 1 128 = 28–1 625 = 56–2 4088 = 84 – 8 – 0
13825 = 1 – (8 – 25)3 15617 = 56 – 71 – 1 15618 = 1 – (81 – 56) 15621 = 56 – 2 – 1 – 1
15622 = 56 – 2 – (2 – 1) 15623 = 56 – (3 – 2) – 1 15624 = 56 – (4 – 2 – 1) 15626 = 56 – (16 – 2)
15632 = 56 – (1 – 23) 15633 = 56 – (1 – 3)3 16374 = 47 – 1 – 6 – 3 16375 = (5 – 1)7 – 6 – 3
16377 = (1 – (3 – 6))7 – 7 16378 = (8 – 3 – 1)7 – 6 16384 = 48–16–3 16743 = 76–1 – 43
16794 = 76–1 – 9 – 4 16798 = 76–18 – 9 16807 = 76–1 – 08 19692 = 9 – (1 – (2 – 6))9
19693 = 1 – ((3 – 6)9 – 9) 32759 = (3 – (2 – 7))5 – 9 32761 = (26–1)3 – 7 32762 = (27–2)3 – 6
32768 = ((8 – 6)3)7–2 32775 = 7 – (2 – 3 – 7)5 43264 = (63 – 4 – 4)2 45632 = (63)2 – 45
46633 = 66 – (33 – 4) 46644 = 66 – 4 – 4 – 4 46648 = (4 – (4 – 6))6 – 8 46650 = 66 – 5 – 40
46651 = 66 – 15 – 4 46653 = 66 – 35–4 46655 = 66 – 45–5 46656 = (66)(5–4)6
46657 = 66 – (4 – 5)7 46660 = 4 – (06 – 66) 46661 = 66 – (14 – 6) 46663 = 66 – (3 – 4 – 6)
54958 = 95 – (84 – 5) 58921 = 95 – 28–1 59038 = 95 – 8 – 3 – 0 59039 = 95 – 9 – 30
59044 = 95 – 4 – 40 59045 = 95 – 4 – 05 59046 = 95 – (4 – 60) 59048 = 95 – (40)8
59050 = 95 – (0 – 50) 59052 = 95 – 2 – (0 – 5) 59053 = 95 – (30 – 5) 59054 = 95 – (04 – 5)
59129 = 95 – (1 – 92) 59263 = 95 – (2 – 63) 59392 = 95 – (2 – 9)3 78115 = 57 – 8 – 1 – 1
78116 = (6 – 1)7 – 8 – 1 78117 = (6 – 1)7 – 81 78123 = (8 – 3)7 – 21 78133 = 8 – (1 – 3 – 3)7
78135 = 8 – (1 – 57 – 3) 78136 = 8 – ((1 – 6)7 – 3) 97966 = 76 – (9 – 6)9 246519 = 49 – 5612 (JD)
248832 = (8–(2–2–4))8–3 (JD) 259743 = 49 – 75–(3–2) (JD) 278640 = 67 – (8 – 2)4 – 0 (JD) 279680 = 67 – 28 – 09 (JD)
279936 = 67 – (9 – 9)3/2 (JD) 524284 = 28–(2–5–8) – 4 (JD) 524288 = 28–(4–5–2–8) (JD) 531441 = 3(51–(1–4–4)) (JD)
1048576 = 45–(8–7–6)–01 (JD) 1679608 = 68 – 7 – 1960 (JD) 1679616 = 6(9–1)1766 (JD) 1946564 = 59 – (4–(16–6))4 (JD)
1950724 = 59 – 74 – 012 (JD) 1952395 = 59 – 93 – 125 (JD) 1952396 = 59 – 93612 (JD) 1952868 = 59 – 28 – 168 (JD)
1952869 = 59 – 28169 (JD) 1952997 = 59 – 27199 (JD) 1953042 = 59 – 34 – 210 (JD) 1953043 = 59 – 34 – 130 (JD)
1953044 = 59 – 34 – 014 (JD) 1953113 = (53)3 – 9 – 1 – 1 – 1 (JD) 1953114 = (53)4–1 – 9 – 1 – 1 (JD) 1953115 = 59 – 5 – 3 – 1 – 11 (JD)
1953116 = 59 – 6 – 3111 (JD) 1953117 = 59 – 7 – 1113 (JD) 1953118 = 59 – (8–1)113 (JD) 1953119 = 59 – 3 – 1 – 1 – 19 (JD)
1953120 = 59 – 31 – 210 (JD) 1953121 = 59 – 3121 – 1 (JD) 1953122 = 59 – 31221 (JD) 1953123 = 59 – 21133 (JD)
1953124 = 59 – 13124 (JD) 2097151 = 2(9–(01–7–5)) – 1 (JD) 2097152 = 2(9–(02–1–7–5)) (JD) 4194300 = 49–(1–3–0) – (4 – 0) (JD)
4194301 = 49–(1–3–0) – (4 – 1) (JD) 4194302 = 49–(1–3–0) – (4 – 2) (JD) 4194303 = 49–(1–3–0) – (4 – 3) (JD) 4194304 = 49–(1–3–0) – (4 – 4) (JD)
4194305 = 49–(1–3–0) – (4 – 5) (JD) 4194306 = 49–(1–3–0) – (4 – 6) (JD) 4194307 = 49–(1–3–0) – (4 – 7) (JD) 4194308 = 49–(1–3–0) – (4 – 8) (JD)
4194309 = 49–(1–3–0) – (4 – 9) (JD) 4782969 = 97(9–8)246 (JD) 8388608 = (8 – 6)8–(30–8–8) (JD)

Exponentiation and Multiplication
2592 = 25 × 92 2916 = (1 × 6 × 9)2 23328 = (2 × 33)2 × 8 28672 = 7 × 26 × 82
32805 = 5 × 38 × 20 34805 = 520 × 38 34992 = 3 × 9 × (4 × 9)2 82944 = (8 × 9)2 × 4 × 4
262144 = (2 × 4)6 × 12×4 (JD) 1594323 = 94 × 35 × 12×3 (BH) 1679616 = 67 × 61 × 16×9 (JD) 14348907 = 37 × 94 × 14×8×0 (BH)

Exponentiation and Division
11664 = 1 / (1 / 66)) / 4 (BH) 46656 = 6(65/64) (JD) 279936 = 67(9/9)3/2 (JD) 2097152 = (27)91/250 (JD) 1679616 = 69 / 61 / 16/7 (JD)
4782969 = 97(8/4/2)69 (JD)

Exponentiation and Concatenation
121 = 112 9216 = (962)1 11025 = (1052)1 13824 = (243)18 19321 = (1392)1
39304 = (343)90 81225 = (2852)1 131072 = 217130 (JD) 161051 = 115160 (BH) 1771561 = 1161577 (BH)
2097152 = 2219750 (JD) 19487171 = 11714789 (BH)

Exponentiation and Decimal Points
2097152 = (27)90.512 (JD)

Roots
0 = √(0) 1 = √(1)

Roots and Exponentiation
64 = √(4)6 2097152 = (27)√91250(JD)

Roots and Decimal Points
1024 = 1√(0.2√4) (JD) 3125 = (3√1)√(.2√(5)) (JD)

Factorial
1 = 1! 2 = 2!

Factorial and Addition
144 = (1 + 4)! + 4! 145 = 1 + 4! + 5! 733 = 7 + 3! + (3!)! 5160 = 5! + (1 + 6)! + 0
5161 = 5! + (1 + 6)! + 1 5162 = 5! + (1 + 6)! + 2 5163 = 5! + (1 + 6)! + 3 5164 = 5! + (1 + 6)! + 4
5165 = 5! + (1 + 6)! + 5 5166 = 5! + (1 + 6)! + 6 5167 = 5! + (1 + 6)! + 7 5168 = 5! + (1 + 6)! + 8
5169 = 5! + (1 + 6)! + 9 40440 = (4 + 0!)! + (4 + 4)! + 0 40441 = (4 + 0!)! + (4 + 4)! + 1 40442 = (4 + 0!)! + (4 + 4)! + 2
40443 = (4 + 0!)! + (4 + 4)! + 3 40444 = (4 + 0!)! + (4 + 4)! + 4 40445 = (4 + 0!)! + (4 + 4)! + 5 40446 = (4 + 0!)! + (4 + 4)! + 6
40447 = (4 + 0!)! + (4 + 4)! + 7 40448 = (4 + 0!)! + (4 + 4)! + 8 40449 = (4 + 0!)! + (4 + 4)! + 9 40584 = (4 + 0!)! + 5! + 8! + 4!
40585 = 4! + 0! + 5! + 8! + 5! 363000 = (3 + 6)! + (3 + 0! + 0!)! + 0 (BH) 363001 = (3 + 6)! + (3 + 0! + 0!)! + 1 (BH) 363002 = (3 + 6)! + (3 + 0! + 0!)! + 2 (BH)
363003 = (3 + 6)! + (3 + 0! + 0!)! + 3 (BH) 363004 = (3 + 6)! + (3 + 0! + 0!)! + 4 (BH) 363005 = (3 + 6)! + (3 + 0! + 0!)! + 5 (BH) 363006 = (3 + 6)! + (3 + 0! + 0!)! + 6 (BH)
363007 = (3 + 6)! + (3 + 0! + 0!)! + 7 (BH) 363008 = (3 + 6)! + (3 + 0! + 0!)! + 8 (BH) 363009 = (3 + 6)! + (3 + 0! + 0!)! + 9 (BH) 363024 = (3 + 6)! + (3 + 0 + 2)! + 4! (BH)
363024 = (3 + 6)! + 3! + 6! + 1 + 3! (BH)

Factorial and Subtraction
715 = (7 – 1)! – 5 720 = (7 – (2 – 0!))! 1435 = (4 – 1)!! – (5 – 3!!) 1463 = 4! – (1 – 6! – 3!!)
4296 = (9 – 2)! – 4! – 6! 4317 = 7! – 3!! – (4! – 1) 5017 = 7! – ((5 – 1)! – 0!) 5035 = ( 5 – (0! – 3))! – 5
5037 = 7! – 0! – (5 – 3) 5038 = (8 – 0)! – (5 – 3) 5039 = (9 – (5 – 3))! – 0! 5040 = (5 – (0! – 4) – 0!)!
5637 = 7! – (5! – 6!) – 3 6476 = 6! – (4 – 7! – 6!) 10077 = -1 – 0! – (0! – 7! – 7!) 10078 = (8 – 0!)! – 0! – (1 – 7!)
10079 = (9 – 0! – 0!)! – (1 – 7!) 35280 = 8! – (5 – (2 – 3 – 0!))! 39600 = (9 – (3 – 0! – 0!))! – 6! 40287 = 8! – 4! – 7 – 2 – 0
40289 = 8! – 4! – (9 – 2 – 0) 40294 = (9 – 0!)! – 4! – (4 – 2)) 40296 = (9 – 0!)! – (6 – (4 – 2))! 40298 = 8! – (9 – 4 – 0!)! – 2
40308 = 8! – 4 – 0! – 3! – 0! 40309 = (9 – 0!)! – 4 – 0! – 3! 40314 = (4 – (1 – 0! – 4))! – 3! 40315 = (5 – (0 – 3))! – 4 – 1
40316 = (6 – (1 – 3))! – 4 – 0 40318 = 4 – 0! – (3! – 1 – 8!) 40319 = (9 – 1)! – (4 – 3 – 0) 40320 = (4 – (0! – 3 – 2))! – 0
40321 = (3! – (2 – 4))! – (0 – 1) 40322 = (3! – (2 – 4))! – (0 – 2) 40323 = (3! – (2 – 4))! – (0 – 3) 40324 = (3! – (2 – 4))! – (0 – 4)
40325 = (3! – (2 – 4))! – (0 – 5) 40326 = (3! – (2 – 4))! – (0 – 6) 40327 = (3! – (2 – 4))! – (0 – 7) 40328 = 4 – (0! – 3 – 2 – 8!)
40329 = (3! – (2 – 4))! – (0 – 9)

Factorial and Multiplication
36 = 3! × 6 40320 = (4 × (0 × 3)! × 2)! × 0! (JD) 82944 = 8 × 2 × 9 × 4! × 4!

Factorial and Division
120960 = 9! / (6/2) / 1 / 0! / 0! (JD) 255024 = 4!! / (5 / ((0/5)! / 2 / 2))!(JD)
362880 = (3 / (2/6))! / (8/8) / 0! (JD) 479001600 = (9! / 7! / 6)! / (0/4)! / 0! / 0! / 0! (JD)

Factorial and Exponentiation
13824 = (4!)3182 (JD)

Multiplication and Concatenation
126 = 6 × 21 153 = 3 × 51 688 = 8 × 86 1206 = 6 × 201 1255 = 5 × 251
1260 = 6 × 210 1395 = 15 × 93 1435 = 35 × 41 1503 = 3 × 501 1530 = 3 × 510
1827 = 21 × 87 2187 = 27 × 81 3159 = 9 × 351 3784 = 8 × 473 6880 = 8 × 860
10251 = 51 × 201 10255 = 5 × 2051 10426 = 26 × 401 10521 = 21 × 501 10525 = 5 × 2105
10575 = 15 × 705 11259 = 9 × 1251 11439 = 9 × 31 × 41 11844 = 84 × 141 11848 = 8 × 1481
12006 = 6 × 2001 12060 = 6 × 2010 12384 = 3 × 4128 12505 = 5 × 2501 12546 = 51 × 246
12550 = 5 × 2510 12595 = 5 × 2519 12600 = 6 × 2100 12762 = 6 × 2127 12768 = 8 × 21 × 76
12798 = 2 × 79 × 81 12843 = 3 × 4281 12955 = 5 × 2591 12964 = 14 × 926 13243 = 41 × 323
13545 = 3 × 4515 13764 = 6 × 31 × 74 13950 = 15 × 930 14035 = 35 × 401 14350 = 35 × 410
15003 = 3 × 5001 15030 = 3 × 5010 15246 = 6 × 2541 15300 = 3 × 5100 15435 = 3 × 5145
15624 = 24 × 651 15795 = 9 × 1755 15975 = 5 × 5 × 9 × 71 16272 = 6 × 2712 16368 = 8 × 31 × 66
17325 = 75 × 231 17346 = 6 × 7 × 413 17428 = 2 × 8714 17437 = 47 × 371 17482 = 2 × 8741
17892 = 9 × 28 × 71 18225 = 81 × 225 18265 = 65 × 281 18270 = 21 × 870 19026 = 21 × 906
19215 = 21 × 915 19845 = 5 × 49 × 81 21375 = 3 × 7125 21586 = 86 × 251 21753 = 3 × 7251
21870 = 27 × 810 24768 = 4 × 72 × 86 25105 = 5 × 5021 25375 = 35 × 725 25474 = 47 × 542
25510 = 5 × 5102 28476 = 42 × 678 29184 = 4 × 8 × 912 29632 = 32 × 926 31509 = 9 × 3501
31590 = 9 × 3510 33579 = 7 × 9 × 533 33655 = 53 × 635 33696 = 36 × 936 34992 = 9 × 9 × 432
36855 = 63 × 585 37668 = 6 × 73 × 86 37840 = 8 × 4730 37845 = 87 × 435 39784 = 8 × 4973
41665 = 65 × 641 42898 = 89 × 482 44676 = 6 × 7446 45684 = 54 × 846 45760 = 65 × 704
45864 = 84 × 546 47538 = 57 × 834 48672 = 78 × 624 49855 = 59 × 845 49896 = 6 × 84 × 99
49968 = 8 × 9 × 694 51759 = 9 × 5751 52168 = 8 × 6521 53865 = 63 × 855 56295 = 9 × 6255
56875 = 65 × 875 62968 = 68 × 926 63895 = 65 × 983 67149 = 9 × 7461 67392 = 72 × 936
67950 = 75 × 906 68800 = 8 × 8600 69984 = 9 × 9 × 864 71199 = 9 × 7911 78975 = 9 × 8775

Fibonacci
0 = F0 1 = F1 5 = F5

Fibonacci and Addition
55 = F5+5 233 = FF(F2+3+3) 234 = F2 + FF3+4 235 = 2 + FF(F3+4) 236 = 3 + FF(F2+6)
237 = 2 + F3 + FF7 267 = F(F2+F6) + FF7 2584 = F(F2+5+8+4) 2585 = F2 + F5+8+5 2589 = F(F2+8+9) + 5
6772 = 6 + F(F7+7) + F2 6773 = 6 + F(F7+7) + F3 6774 = 6 + F(F7+7) + F4

Fibonacci and Subtraction
376 = F(7–FF6) – FF3 377 = F(FF(7–FF3)–7) 1576 = F(F7–(1–5)) – FF6 1596 = F(FF6–(9–5)) – 1
1597 = F(F9–F7–(5–1)) 2582 = F(2–(5–F8)) – 2 2583 = F(F3–(5–F8)) – F2 2584 = F(F8–2–(5–4))
4177 = F(F7–(1–7)) – 4 4180 = F(F8–FF4) – 1 – 0 4181 = F(4–(1–F8–1)) 6763 = F(FF6–(7–6)) – F3 (BH)
6764 = F(FF6–(7–6)) – FFF4 (BH) 6765 = F(F(FF6–F7)–(6–5)) 10857 = FF8 – F0–(1–5–7) (BH) 10943 = FFF1–(0–9)–4 – 3 (BH)
10944 = FFF1–(0–9)–4 – FF4 (BH) 10946 = FF1–(0–9–4)–6 (BH) 17711 = F(F7–7) – (1–1–1) (BH) 28657 = F(2–(8–FF6)–(5–F7)) (BH)

Fibonacci and Multiplication
34 = F(3×F4) 63 = FF6 × 3 84 = F8 × 4 144 = 1 × F(F4×4) 168 = 1 × F6 × F8
189 = 1 × F8 × 9 273 = FF2×3 × F7 495 = F(FF4×5) × 9 882 = F8 × F8 × 2 1296 = 1 × F2×6 × 9
2646 = 2 × FF6 × F4 × FF6 2688 = 2 × F6 × F8 × 8

Lucas
1 = L1

Lucas and Addition
32 = 3 + LLLL2 58 = L5 + L8 123 = L(1+2+LL3) 124 = 1 + L(L2+L4) 199 = L1+9 + L9
201 = 2 + LL(LLL0+1) 202 = L2 + LL(L0+L2) 203 = LL2 + LL(L0+3) 205 = LL2 + L0 + LL5 228 = LLLL2 + L(L2+8)
235 = LLL2 + LLL3 + LL5 257 = LLLL2 + LL5 + L7 275 = L2+7 + LL5 322 = L(3+2+LLL2) 325 = 3 + L(LLL2+5)
329 = LL3 + L(L2+9) 351 = LLL3 + L(L5+1) 398 = L9 + L(L3+8) 521 = L(5+LLL2+1) 526 = 5 + L(LLL2+6)
528 = LLL2 + L5+8

Lucas and Subtraction
22 = LLLL2 – LLL2 27 = L7 – 2 29 = LLLL(LLL2–9) 123 = L(L(LL3–2)–1) 521 = L(L(LLL2–1)–5) 843 = L(L8–4–LLL3)

Lucas and Multiplication
35 = LL3 × 5 55 = L5 × 5 126 = 1 × LLL2 × L6 145 = LL1×4 × 5 203 = L2×L0 × LLL3
232 = 2 × L3 × LLLL2 235 = L(2×L3) × 5 245 = LLL2 × L4 × 5 282 = 2 × L8 × L2 304 = L(3×LL0) × 4
322 = L3×2×2 343 = LL3 × L4 × LL3 435 = LL4 × 3 × 5

Binomial
28 = 8C2 (ZA)

Binomial and Addition
6435 = 6C4C3+5 6438 = 6C4C8 + 3 18564 = 1+8+5+4C6 18565 = 8+5+5C6 + 1 18569 = 1+8+9C6 + 5
20475 = (7+5C0)C2C4 26235 = 5+6C2C3C2 75582 = 7+5+5+2C8 75587 = 7+5+7C8 + 5 75589 = 5+5+9C8 + 7
92378 = 3+7+9C2+8 118755 = (8C1+7C5)C5C1 118756 = (1+8C7C6)C5 + 1 122265 = (1+6+5C2+2)C2 122266 = (6+6C2+2)C2 + 1
123753 = (7+5C1+3+3)C2 123754 = (7+5C4+3)C2 + 1 134596 = (5C3+1+4+9)C6 148995 = (1+5C9C9)C8C4 167960 = 1+6+7+6C9 + 0
167961 = 1+6+7+6C9 + 1 167962 = 1+6+7+6C9 + 2 167963 = 1+6+7+6C9 + 3 167964 = 1+6+7+6C9 + 4 167965 = 1+6+7+6C9 + 5
167966 = 1+6+7+6C9 + 6 167967 = 1+6+7+6C9 + 7 167968 = 1+6+7+6C9 + 8 167969 = 1+6+7+6C9 + 9 184756 = 8+7+5C(4+6C1)

Binomial and Subtraction
4996 = 6C4C9 – 9 7139 = 9C7C3 – 1 12376 = (7C2–1–3)C6 38760 = 6C3C(7–8C0) 43758 = (7C5C4–3)C8
46376 = (7C3–6C6)C4 54738 = 8C4C3 – (7 – 5) 77515 = (7C5–1)C7 – 5 77520 = (7C5–2C0)C7 92375 = (7C5–2)C9 – 3
92378 = (7C8–3–2)C9 95284 = 9C4C2C3 98265 = 8C2C5 – 6 – 9 116277 = 7C6–1C7 – 1 – 2 116278 = 8–1C6–1C7 – 2
116279 = 9–2C6–1C7 – 1 125963 = 6C3C9–1 – 2 – 5 125965 = 6C5–2C9–1 – 5 125968 = 6C8–5C9–1 – 2 125970 = (7C2–5C0)C9–1
134596 = (9–1C5–3–4)C6 167945 = (7C5–1)C9 – 6C4 167952 = (7C5–1)C9 – 6 – 2 167953 = 6C5–(3–1)C9 – 7 167954 = 6C7–4C9 – 1 – 5
167955 = 6C7–(5–1)C9 – 5 167958 = (7C5–1)C9 – (8 – 6) 167959 = 6C7–(9–5)C9 – 1 167960 = (7C6–1–6C0)C9 194580 = (8C5–9)C4 – 1C0
203487 = 7C2C8 – 3C4C0

Binomial and Multiplication
2268 = 6 × 8C2C2 2376 = 3 × 2×6C7 15504 = (5×4×1C0)C5 17892 = 1 × 7 × 8×9C2 22896 = 2 × 8 × 6×9C2
24384 = 3 × 4×4×8C2 24864 = 4 × (4×8C6)C2 32725 = 5 × (7C2×2)C3 36288 = 3 × 6 × 8×8C2 39711 = 1 × 1 × 9×7C3
74382 = 7 × (8×3C2)C4 82368 = 8 × 8×6×3C2 92664 = 9 × 6×6×4C2 123984 = 1 × 3 × 9×8×4C2 139536 = 1 × 9 × 6C3C3×5
142506 = 1C0 × 2×6C4C5 174832 = 1 × 7 × 4×8C3C2 198432 = 1 × 8 × 9×4C2C3

Binomial and Fibonacci
56 = F6C5 (ZA)

#2.

Almost Friedman Numbers (written with the digits of n–1)
128 = 1 × 27 1280 = (1 + 9) × 27 1295 = (4 × 9)2 – 1 1330 = (2 + 9)3 – 1 1365 = (46 – 1) / 3
1763 = -1 + (7 × 6)2 1764 = (6 × 7)3–1 2048 = (2 + 0)4+7 2179 = (2 + 1)7 – 8 2187 = (6 / 2)8–1
2688 = 82 × 6 × 7 2744 = (7 × 4 / 2)3 2916 = (9 × (5 + 1))2 2917 = (9 × 6)2 + 1 3215 = (3 + 2)1+4
3126 = (3 + 2)5 + 1 3579 = 83 × 7 – 5 3600 = (93 – 9) × 5 3936 = (39 – 3) / 5 4096 = (9 – 50)4
4097 = 46 + 90 4370 = 93 × 6 – 4 4793 = 74 × 2 – 9 5282 = 31+5 × 8 5837 = 5 + 8 × 36
5929 = (9 × 8 + 5)2 6256 = 6 + 2 × 55 6550 = 94 – 6 – 5 7776 = (7 – 7 / 7)5 10239 = 20 × 83 – 1
10240 = 20 × (9 – 1)3 10353 = 203 × 51 10626 = 506 × 21 11025 = (101 + 4)2 11026 = 1052 + 1
11236 = ((1 + 1) × 53)2 11664 = 1 × 66 / (1 + 3) 11665 = 1 + 1 × 66 / 4 12264 = 6132 × 2 12288 = 12 × 8 × 27
12347 = 423 / 6 – 1 12348 = 423 / (7 – 1) 12540 = 132 × 95 12544 = (14 × (3 + 5))2 12549 = (14 × 8)2 + 5
12624 = 6312 × 2 12636 = 52 × 36–1 12696 = 6 × (5 × 9 + 1)2 12800 = (99 + 1) × 27 12996 = (99 + 15)2
13225 = (23 × (1 + 4))2 13226 = (23 × 5)2 + 1 13593 = 591 × 23 13593 = 217 × 63 13699 = 19 × (36 – 8)
13755 = 5 × (143 + 7) 13824 = (2 – 1) × (3 × 8)3 13825 = (2 × (4 + 8))3 + 1 13888 = 31 × 8 × 8 × 7 13924 = (39 × 3 + 1)2
14028 = 7014 × 2 14208 = 7104 × 2 14298 = 7149 × 2 14632 = 114 – 6 – 3 14635 = (3 × 4 – 1)4 – 6
14638 = (17 – 6)4 – 3 14640 = (6 / 3 + 9)4 – 1 14641 = (1 + 4 + 6)4 + 0 14642 = (1 + 4 + 6)4 + 1 14643 = (1 + 4 + 6)4 + 2
14644 = (1 + 4 + 6)4 + 3 14645 = (1 + 4 + 6)4 + 4 14646 = (1 + 4 + 6)4 + 5 14647 = (1 + 4 + 6)4 + 6 14648 = (1 + 4 + 6)4 + 7
14649 = (1 + 4 + 6)4 + 8 14650 = (1 + 4 + 6)4 + 9 14651 = (5 + 6)4 + 10 14652 = (5 + 6)4 + 11 14653 = (5 + 6)4 + 12
14654 = (5 + 6)4 + 13 14655 = (5 + 6)4 + 14 14656 = (5 + 6)4 + 15 14657 = (5 + 6)4 + 16 14658 = (5 + 6)4 + 17
14659 = (5 + 6)4 + 18 14660 = (5 + 6)4 + 19 14688 = 864 × 17 14711 = 114 + 70 14712 = 114 + 71
14713 = 114 + 72 14714 = 114 + 73 14715 = 114 + 74 14716 = 114 + 75 14717 = 114 + 76
14718 = 114 + 77 14719 = 114 + 78 14720 = 114 + 79 15129 = (115 + 8)2 15342 = 5114 × 3
15345 = 341 × 45 15355 = 5 × (3 × 45 – 1) 15369 = 56 – (3 – 1)8 15374 = 7 × 133 – 5 15375 = 375 × 41
15376 = (53 – 1)7–5 15379 = 1 × 7 × (5 + 8)3 15435 = 441 × 35 15444 = 55+1 – 34 15555 = (55 – 14) × 5
15562 = (65 + 5) × (1 + 1) 15563 = (65 + 5) × 2 + 1 15564 = 52×3 – 61 15565 = 56 – 15 × 4 15568 = 56 – 57 × 1
15570 = 591 × 5 × 6 15573 = 57–1 – 52 15585 = (1 + 4) × (55 – 8) 15586 = 1 + (55 – 8) × 5 15608 = 56 – 17 + 0
15613 = 1 × 56 – 12 15617 = -1 + 56 – 1 – 6 15618 = 1 × 56 – 1 × 7 15619 = 1 + 56 + 1 – 8 15622 = 1 × 56 – 2 – 1
15623 = -1 + 56 – 2 / 2 15624 = 1 × 56 + 2 – 3 15625 = 56 × 124 15626 = 56 + 125 15627 = 56 + 6 / (1 + 2)
15628 = 56 + 2 + 17 15629 = 56 + 12 – 8 15630 = 56 + (1 + 9) / 2 15633 = -1 + 56 + 32 15634 = 1 × 56 + 3 × 3
15637 = 56 + 6 × (3 – 1) 15645 = 56 + 4 × (4 + 1) 15646 = 1 + 56 + 4 × 5 15649 = 56 + 8 × (4 – 1) 15655 = 5 × (51+4 + 6)
15656 = 1 + 5 × (6 + 55) 15667 = 56 + 6 × (6 + 1) 15668 = 56 + 6 × 7 + 1 15670 = 56 + 9 × (6 – 1) 15673 = -1 + 56 + 72
15674 = 56 + 73–1 15688 = 56 + 7 × (8 + 1) 15689 = 1 × 56 + 8 × 8 15694 = 51+5 + 69 15735 = 3147 × 5
15753 = 51+5 + 27 15988 = 75 – 891 16129 = (16 × 8 – 1)2 16224 = 3122 / 6 16296 = 291 × 56
16348 = 47 – 36 × 1 16375 = 47 – 9 × 16 16378 = (7 – 3)7 – 6 × 1 16383 = 26+8 – 13 16384 = (3 – 136+8
16385 = 85 × 3 / 6 + 1 16387 = 86 / 16 + 3 16448 = 47 + 64 × 1 16464 = 146–3 × 6 16480 = 47 + 96 × 1
16498 = 47 + 6 × 19 16524 = 51 × (3 × 6)2 16743 = 75 – 26 16744 = 75 – 63 16754 = 76–1 – 53
16758 = (1 + 6)5 – 7 × 7 16759 = 75 – 6 × 8 × 1 16765 = (74 – 6) × (6 + 1) 16790 = 76–1 – 8 – 9 16798 = 76 / 7 – 9 × 1
16799 = 76–19 – 8 16806 = (6 + 80)5 – 1 16807 = (6 + 80)6–1 16808 = 76–1 + 80 16815 = (6 + 1)4+1 + 8
16816 = (6 + 1)5 + 8 + 1 16848 = 78 × 64–1 16854 = 5618 × 3 16855 = (6 + 1)5 + 48 16875 = 74+1 + 68
16876 = 1 + 68 + 75 16896 = (6 + 1)5 + 89 16976 = 169 + 75 17327 = 3 × 762 – 1 17328 = 3 × (77 – 1)2
17355 = 3471 × 5 17423 = (27 + 4)2 – 1 17424 = (7 + 4)3–1 17484 = 8 × (37 – 1) – 4 17488 = 8 × ((7 – 4)7 – 1)
17496 = 9 × (7 – 1)5 / 4 17508 = 75 + 701 17518 = 75 + 711 17528 = 75 + 721 17536 = 75 + 31+5
17537 = 75 + 36 + 1 17538 = 75 + 731 17548 = 75 + 741 17558 = 75 + 751 17568 = 75 + 761
17578 = 75 + 771 17588 = 75 + 781 17598 = 75 + 791 17733 = 771 × 23 17857 = (8 × 56 – 1) / 7
18480 = 8 × (74 – 91) 18944 = (4 × 9 + 1) × 83 18954 = 81 × (35 – 9) 19321 = 1392 + 0 19322 = 1392 + 1
19323 = 1392 + 2 19324 = 1392 + 3 19325 = 1392 + 4 19326 = 1392 + 5 19327 = 1392 + 6
19328 = 1392 + 7 19329 = 1392 + 8 19330 = 1392 + 9 19638 = 9 × (37 – 6 + 1) 19647 = (4 – 1)9 – 62
19653 = (1 + 2)9 – 6 × 5 19654 = 39 – 6 × 5 + 1 19674 = 9 × (37 – 16) 19682 = (8 + 1 – 6)9 – 1 19683 = (8 + 2 – 6 – 1)9
19684 = 39 + 16+8 19685 = (4 – 1)9 + 8 – 6 19690 = (9 – 6)9 + 8 – 1 19692 = (6 / (1 + 1))9 + 9 19693 = (6 / 2)9 + 9 + 1
19732 = 39 + 71–1 19733 = 39 + 72 + 1 19737 = 1 × 9 × (37 + 6) 19739 = 39 + 7 × 8 × 1 19764 = 9 × ((6 + 7)3 – 1)
29525 = (2 × 95 + 2) / 4 (BH) 29526 = (95 + 5 – 2) / 2 (BH) 29528 = 95 / 2 + 7 / 2 (BH) 29530 = (95 + 9 + 2) / 2 (BH)

Almost Friedman Numbers (written with the digits of n+1)
1225 = (62 – 1)2 1372 = (1 + 3) × 73 1373 = 4 × 73 + 1 1555 = (65 – 1) / 5 1935 = (63 – 1) × 9
2048 = 29 × 4 + 0 2178 = (2 + 1)7 – 9 2196 = (2 + 1)7 + 9 2500 = 502 × 1 2915 = (9 × 6)2 – 1
2916 = (9 × (7 – 1))2 3124 = (3 + 2)5 – 1 3125 = (3 + 2)6–1 3277 = (38 – 7) / 2 3280 = (38 – 1) / 2
3282 = (38 + 3) / 2 3375 = (3 × 7 – 6)3 3650 = (36 + 1) × 5 3773 = 73 × (4 + 7) 3853 = 84 – 35
4095 = 46 – 90 4096 = (7 + 90)4 4097 = 84 + 90 5832 = 35 × 3 × 8 6144 = 6 × 1 × 45
6250 = (6 – 1)5 × 2 6666 = (66 + 6) / 7 10570 = 151 × 70 10860 = 60 × 181 10920 = 12 × 910
11024 = 1052 – 1 11025 = (106 – 1)2 11236 = (113 – 7)2 11264 = 11 × (-2 + 6)5 11628 = 612 × 19
11663 = -1 + 1 × 66 / 4 11664 = 1 × 66 / (5 – 1) 11767 = 1681 × 7 11934 = 9 × (113 – 5) 11972 = 9 × 113 – 7
12062 = 6031 × 2 12100 = 1 × 1102 12288 = 192 × 82 12348 = 9 × 143 / 2 12393 = (24 + 1) × 93
12444 = 244 × 51 12444 = 271 × 46 12488 = 892 × 14 12505 = 205 × 61 12544 = 4 × (55 + 1)2
12545 = 4 × 562 + 1 12586 = 217 × 58 12597 = (28 – 9) × 51 12602 = 6301 × 2 12638 = 6319 × 2
12768 = 672 × 19 12800 = (1 + 1) × 802 12801 = 1 + 2 × 802 12816 = 712 × 18 12874 = 157 × 82
12978 = 721 × (9 + 9) 13122 = 1 × 2 × 323 13224 = (23 × 5)2 – 1 13225 = (23 × (6 – 1))2 13286 = 73 × 182
13377 = (31 + 8) × 73 13378 = 39 × 73 + 1 13503 = 4501 × 3 13797 = 891 × 73 13822 = (3 × 8)3 – 1 × 2
13823 = (2 × (4 + 8))3 – 1 13824 = 1 × (3 × 8)-2+5 13825 = (6 × 8 / 2)3 + 1 13855 = 163 × 85 13924 = (59 × (3 – 1))2
13946 = 734 × 19 13965 = 19 × (36 + 6) 13986 = 518 × 9 × 3 14194 = 151 × 94 14375 = 6 × 74 – 31
14406 = (10 – 4) × 74 14640 = (1 + 4 + 6)4 – 1 14641 = (2 + 4 + 6 – 1)4 14644 = (5 + 6)4 + 4 – 1 14645 = (6 + 6 – 1)4 + 4
14646 = (4 + 7)4 + 6 – 1 14647 = (4 + 8 – 1)4 + 6 14758 = (95 – 17) / 4 14872 = 3714 × 4 14896 = 784 × 19
15025 = 601 × 25 15042 = 5014 × 3 15305 = 3061 × 5 15325 = 613 × 25 15386 = 7 × ((5 + 8)3 + 1)
15435 = (6 + 1)3 × 45 15436 = 73 × 45 + 1 15525 = 621 × 5 × 5 15552 = (1 + 5)5 × (5 – 3) 15561 = (65 + 5) × 2 – 1
15562 = (65 + 5) × (3 – 1) 15564 = 5 × 56 – 61 15567 = 56 – 58 × 1 15572 = 57–1 – 53 15574 = 57 / 5 – 51
15576 = 51+5 – 7 × 7 15580 = 57–1 – 5 × 9 15584 = -1 + (55 – 8) × 5 15585 = 56 – 5 × 8 × 1 15607 = 56 – 18 + 0
15612 = 1 × 56 – 13 15616 = -1 + 56 – 1 – 7 15617 = 1 × 56 – 1 × 8 15618 = 1 + 56 – 1 + 9 15621 = 1 × 56 – 2 – 2
15622 = 56 – 3 × 12 15623 = 1 × 56 + 2 – 4 15624 = 56 – 125 15625 = 56 × 126 15626 = 56 + 127
15627 = 56 + 2 × 18 15628 = 56 + 2 + 19 15629 = 1 + 56 + 3 + 0 15630 = 1 + 56 + 3 + 1 15631 = 1 + 56 + 3 + 2
15632 = 1 + 56 + 3 + 3 15633 = 1 + 56 + 3 + 4 15634 = 1 + 56 + 3 + 5 15635 = 1 + 56 + 3 + 6 15636 = 1 + 56 + 3 + 7
15637 = 1 + 56 + 3 + 8 15638 = 1 + 56 + 3 + 9 15639 = 56 + 14 + 0 15640 = 56 + 14 + 1 15641 = 56 + 14 + 2
15642 = 56 + 14 + 3 15643 = 56 + 14 + 4 15644 = 56 + 14 + 5 15645 = 56 + 14 + 6 15646 = 56 + 14 + 7
15647 = 56 + 14 + 8 15648 = 56 + 14 + 9 15650 = 56 + 51+1 15651 = 1 + 56 + 52 15654 = -1 + 5 × (6 + 55)
15655 = 1 × 5 × 6 + 56) 15657 = 56 + 8 × (5 – 1) 15661 = 1 × 56 + 62 15662 = 56 + 36 + 1 15666 = 56 + 6 × 7 – 1
15667 = 56 + 6 × (8 – 1) 15673 = 56 + 47 + 1 15683 = (1 + 4)6 + 58 15684 = 56 + 58 + 1 15688 = 56 + 9 × (8 – 1)
15695 = 56 + 69 + 1 15697 = 1 × 56 + 9 × 8 15762 = 56 + 137 15793 = 47 – 591 15805 = 56 + 180
15871 = (18 × 7)2 – 5 15872 = (5 – 1)7 – 83 15876 = (18 × 7)7–5 15906 = 75 – 901 16206 = 2701 × 6
16224 = 522 × 6 × 1 16348 = (46 – 9) × (1 + 3) 16354 = 56 + 31+5 16355 = 56 + 36 + 1 16356 = 56 + 731
16371 = (6 – 2)7 – 13 16377 = (3 – 1)6+8 – 7 16378 = (1 + 9 / 3)7 – 6 16381 = 26+8 – 3 × 1 16383 = 86 – 4/3 – 1
16384 = 1 / 6 × 3 × 85 16385 = (6 / 3)6+8 + 1 16386 = (1 + 3)7 + 8 – 6 16393 = 46 × (1 + 3) + 9 16397 = 863 × 19
16426 = 214 + 6 × 7 16434 = (143 – 5) × 6 16444 = (46 + 15) × 4 16445 = 4 × 46 + 61 16474 = 47 + 6 × 15
16525 = 661 × 25 16538 = 56 + 913 16596 = 56 + 971 16632 = 33 × 616 16734 = (1 + 6)5 – 73
16743 = 71+4 – 64 16744 = 75 – 64 + 1 16746 = 7 × 74 – 61 16753 = 76–1 – 54 16756 = 76 / 7 – 51
16794 = 75 – 19 + 6 16798 = 76 – 19 – 9 16806 = 76–1 – 80 16807 = (8 – 80)6–1 16813 = (8 – 1)4+1 + 6
16814 = 1 + 6 + (8 – 1)5 16815 = (6 + 1)6–1 + 8 16816 = 76–1 + 8 + 1 16853 = (8 – 1)5 + 46 16855 = (6 + 1)5 + 6 × 8
16856 = 75 + 6 × 8 + 1 16871 = 76–1 + 82 16874 = -1 + 68 + 75 16875 = 76–1 + 68 16896 = 76–1 + 89
17325 = 3 × (762 – 1) 17424 = 4 × (71 – 5)2 17482 = 8 × 37 – 14 17492 = (9 – 1) × 37 – 4 17496 = (9 – 1) × (7 – 4)7
17500 = 7 × 501+1 17501 = 7 × 502 + 1 17535 = -1 + 75 + 36 17536 = (1 + 7) × (5 + 37) 17537 = 8 × (5 + 37) + 1
17649 = 105 – 76 17848 = 184 × 97 17952 = (73 + 9) × 51 17993 = 947 × 19 18225 = (2 × 68 – 1)2
18414 = 18 × (45 – 1) 18450 = 18 × (45 + 1) 18538 = 598 × 31 18662 = 86 × (63 + 1) 19262 = 9631 × 2
19320 = (139 – 1)2 19342 = 39 – 341 19386 = 718 × 9 × 3 19427 = (4 – 1)9 – 28 19638 = 39 – 9 × (6 – 1)
19652 = 39 – 6 × 5 – 1 19653 = (4 – 1)9 – 6 × 5 19681 = (8 + 1 – 6)9 – 2 19682 = 39 – 16+8 19683 = (4 – 16+8)9
19684 = (5 + 6 – 8)9 + 1 19691 = (6 / 2)9 + 9 – 1 19692 = 39 + 9 × 16 19725 = (1 + 2)9 + 6 × 7 19731 = 39 + 72 – 1
19732 = 39 + 73–1 19746 = 9 × ((4 – 1)7 + 7) 29526 = (95 + 7) / 2 – 2(BH) 29528 = (95 + 9 – 2) / 2 (BH) 29538 = (95 + 3 × 9) / 2 (BH)

Bryce Herdt suggested we look for fractional Friedman numbers, integers n whose digits can create some non-integer within 1 of n.

Fractional Friedman Numbers (rounding down)
1364 ≈ 46 / 3 – 1 1365 ≈ (5 – 1)6 / 3 1448 ≈ 814/4 1555 ≈ (1 + 5)5 / 5 1556 ≈ 65 / 5 + 1
1562 ≈ (6 – 1)5 / 2 1674 ≈ 64 + 1/7 1845 ≈ (18 / 4)5 3280 ≈ 38 / 2 + 0 3281 ≈ 38 / 2 + 1
3282 ≈ 38 / 2 + 2 3283 ≈ 38 / 2 + 3 3284 ≈ 38 / 2 + 4 3285 ≈ 38 / 2 + 5 3286 ≈ 38 / 2 + 6
3287 ≈ 38 / 2 + 7 3288 ≈ 38 / 2 + 8 3289 ≈ 38 / 2 + 9 3544 ≈ (5 / 3)4×4 6285 ≈ (6 – 2/8)5
6997 ≈ 967/9 7498 ≈ 8 × 94 / 7 10697 ≈ 1067/9 11663 ≈ (66 – 1) / (3 + 1) 12467 ≈ 4761/2
12687 ≈ (2 + 6/7)+18 12694 ≈ 4 × (69/2 – 1) 12703 ≈ (70 / 3)1+2 12867 ≈ (8 – 1)7 / 26 13178 ≈ 731/8 + 1
13467 ≈ 1437/6 13647 ≈ 34×(1+7/6) 13724 ≈ 134 – 2/7 13823 ≈ (3 × 8)3 – 1 / 2 13824 ≈ 243 + 1 / 8
13985 ≈ 91+5 / 38 13986 ≈ 96 / 38 + 1 14273 ≈ 134 / 2 – 7 14376 ≈ 316/7 – 4 14641 ≈ 114 + 4/6
14755 ≈ 547/5 – 1 14759 ≈ (95 – 7) / 4 – 1 14762 ≈ (7 + 2)6–1 / 4 (BH) 14768 ≈ 4 × 86 / 71 14769 ≈ 96–1 / 4 + 7
14987 ≈ 184 / / – 9 14996 ≈ (2 × 9)4 / (1 + 6) 14997 ≈ (9 + 9)4 / 7 + 1 15216 ≈ (2 × 6 / 5)11 15572 ≈ 75 – 2/51
15622 ≈ 56 – 2 – 1/2 15623 ≈ -1 + 56 – 2/3 15624 ≈ -1 + 56 + 2/4 15625 ≈ 1 × 56 + 2/5 15626 ≈ 1 + 56 + 2/6
15627 ≈ -1 + 56 + 7/2 15628 ≈ 56 + (8 – 1) / 2 15629 ≈ 56 + 9 × 1 / 2 15747 ≈ 47 – 1/(5×7) 15793 ≈ 39 – 1/5 – 7
15860 ≈ (5 + 1/80)6 15903 ≈ (90 / 13)5 15906 ≈ 56 + 1/90 15962 ≈ (6 × (9 – 1))5/2 16258 ≈ 56 + 2/81
16345 ≈ 4 × 531/6 16382 ≈ (86–1 – 3) / 2 16384 ≈ 48–1 + 3/6 16389 ≈ 39 – 18/6 16432 ≈ 3142 / 6
16474 ≈ 6441/7 16497 ≈ (6 × 17 / 9)4 16547 ≈ (7 – 1/46)5 16742 ≈ 47 + 1/26 16759 ≈ 69×5/7 – 1
16797 ≈ (76 – 1) / 7 – 9 16834 ≈ 1 × (6 / 4)3×8 17247 ≈ 47 + 1/27 17542 ≈ (4 + 2/51)7 17567 ≈ (5 + 7/71)6
17689 ≈ 961 + 8/7 17856 ≈ 8 × 56 / 7 – 1 17857 ≈ 8 × 57–1 / 7 17865 ≈ 8 × (56 / 7 + 1) 17928 ≈ (9 / 71/2)8
18183 ≈ (8/3)1+1+8 (BH) 18532 ≈ 183 + 2/5 18592 ≈ 58 / 21 – 9 18599 ≈ (9 / 81/9)5 18724 ≈ 1 × 2 × 48 / 7
19175 ≈ (5 × 9 / 11)7 19242 ≈ (12 – 2/9)4 19483 ≈ 419 × 3 / 8 19682 ≈ (1 + 2)9 – 8/6 19683 ≈ 39 + 1 – 6/8
19684 ≈ (4 – 1)9 + 8/6 19813 ≈ 1 × (31 / 9)8 19847 ≈ (4 + 1 – 8/9)7 19966 ≈ 66 – 9/19 19972 ≈ 2(99+1)/7
19984 ≈ 49 / 81/9 21845 ≈ 2 × 1 × 48 / 6 (BH) 29516 ≈ (95 – 16) / 2 (BH) 29519 ≈ (95 – 1 – 9) / 2 (BH) 29522 ≈ (95 – 2 – 2) / 2 (BH)
29523 ≈ 2 + 95 / 2 – 3 (BH) 29525 ≈ (95 + 5/2) / 2 (BH) 29526 ≈ (95 + 6 – 2) / 2 (BH) 29548 ≈ (95 + 48) / 2 (BH)

Fractional Friedman Numbers (rounding up)
377 ≈ (7 / 3)7 1263 ≈ 36 + 1/2 1556 ≈ 1 × 65 / 5 2654 ≈ 64 + 2/5 3281 ≈ 1 × 38 / 2
3282 ≈ (2 + 1 × 38) / 2 3846 ≈ (63 / 8)4 4666 ≈ 66 / (4 + 6) 4921 ≈ (1 + 2)9 / 4 4923 ≈ 39 / 4 + 2
5344 ≈ 54×4 / 3 5462 ≈ (2 × 4)5 / 6 5642 ≈ (52 / 6)4 6755 ≈ (5 × 7 / 6)5 6945 ≈ 4 × 56 / 9
8296 ≈ (9 / 2)6 – 8 8374 ≈ 487/3 8545 ≈ 545/8 9463 ≈ 39–4/6 (BH) 11163 ≈ ((1+6) / 3)11
11223 ≈ (1 + 1/2)23 11365 ≈ 315/6 + 1 11486 ≈ ((11+8) / 4)6 11664 ≈ (-11 + 66) / 4 11665 ≈ (66 + 1) / (5 – 1)
11765 ≈ 76 / 5 / (1 + 1) 11768 ≈ (17 / 6)1+8 11769 ≈ 1 + (17 / 6)9 12448 ≈ (2 + (1 + 4) / 4)8 12484 ≈ 4 × 48 / 21
12589 ≈ 8 × 195/2 12725 ≈ (1 + 5)7 / 22 12726 ≈ 67 / 22 + 1 12783 ≈ 713 / 28 12937 ≈ 1972 / 3
12945 ≈ (1 × 45 / 9)2 12964 ≈ 96 / 41 + 2 13549 ≈ (3 – 5/41)9 13648 ≈ 1 × 38 + 4/6 13752 ≈ 135 / 27
13778 ≈ (7 – 3)7 – 1/8 13986 ≈ 1× 96 / 38 14315 ≈ 453 / 11 14565 ≈ (4 × 5)16/5 14641 ≈ 114 – 4/6
14742 ≈ 2417/4 14771 ≈ (4 – 1/17)7 14786 ≈ 47 – 6/81 14997 ≈ 94 × (1 + 9/7) 15253 ≈ (-1/5 + 25)3
15259 ≈ (-1 + 5) / (2 / 5)9 15623 ≈ -1 + 56 – 3/2 15624 ≈ -1 + 56 – 2/4 15625 ≈ 1 × 56 – 2/5 15626 ≈ 1 × 56 + 2/6
15627 ≈ 1 + 56 + 2/7 15628 ≈ 56 + 2 + 1/8 15629 ≈ 56 + 9/2 – 1 15632 ≈ 56 + 13/2 15637 ≈ 56 + 1/37
15867 ≈ (5 + 1/78)6 16288 ≈ (2 – 1/6)2×8 16327 ≈ 631/2 / 7 16384 ≈ 48–1 – 3/6 16385 ≈ 3 × (85 + 1) / 6
16390 ≈ 39 – 1/6 + 0 16391 ≈ 39 – 1/6 + 1 16392 ≈ 39 – 1/6 + 2 16393 ≈ 39 – 1/6 + 3 16394 ≈ 39 – 1/6 + 4
16395 ≈ 39 – 1/6 + 5 16396 ≈ 39 – 1/6 + 6 16397 ≈ 39 – 1/6 + 7 16398 ≈ 39 – 1/6 + 8 16399 ≈ 39 – 1/6 + 9
16498 ≈ (1 + 9 + 8/6)4 16582 ≈ 516/2 / 8 16583 ≈ (513 + 6) / 8 16745 ≈ (6 / 51/4)7 16834 ≈ (6 / 4)3×8 – 1
16835 ≈ (6 / (5 – 1))3×8 17134 ≈ 47 + 1/31 17257 ≈ (7 + 1/27)5 17491 ≈ 1 × (7 + 9 / 2)4 17492 ≈ (7 + 9 / 2)4 + 1
17528 ≈ (7 / 2)8 – 1/5 17854 ≈ (17 / 5)8 – 4 17858 ≈ (51 / (7 + 8))8 17866 ≈ 86 – 61/7 17895 ≈ (18 – 7)5 / 9
18276 ≈ (7 / 2)8 – 1/6 18574 ≈ (41 / 8)7 / 5 18724 ≈ 2 × 48 / 7 – 1 18725 ≈ 2 × (5 – 1)8 / 7 18726 ≈ 86 / (2 × 7) + 1
19307 ≈ 103 + 9/7 19368 ≈ 39 – 1/68 19373 ≈ (3 × 9 – 1/7)3 19382 ≈ ((2 + 9) / 8)31 19482 ≈ 819/4 – 2
19484 ≈ 41 + 49/8 19532 ≈ 59–2 / (3 + 1) (BH) 19643 ≈ (93 – 1)6/4 19682 ≈ (6 / 2)9 – 1/8 19683 ≈ 39 + 1 – 8/6
19684 ≈ (4 – 1)9 + 6/8 19693 ≈ 39 + 9 + 1/6 19813 ≈ (31 / 9)8 – 1 29525 ≈ 95 / 2 + 2 / 5 (BH) 29528 ≈ (95 + 8 – 2) / 2 (BH)
29550 ≈ (95 + 50) / 2 (BH)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/28/12.