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 a related question with what we will generally call self-reproducing numbers. 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?

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

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

 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!

 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) 363613 = (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

 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

 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)