Time to differentiate between narcissistic numbers, d-powerful numbers and Friedman numbers. I've mentioned all these before in posts titled Forming Equations from Integers, Nude Numbers, SOD ET AL and Selfie Numbers. This post gathers disparate information into the one place and includes additional information.
NARCISSISTIC NUMBERS
A number $n$ of $k$ digits is called narcissistic if it is equal to the sum of the $k$-th powers of its digits. An example is 9474 where:$$9474=9^4+4^4+7^4+4^4$$These numbers are quite rare and up to 40000 the only ones are 153, 370, 371, 407, 1634, 8208 and 9474. These numbers comprise OEIS A005188:
A005188: Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: \(m\)-digit nonnegative numbers equal to sum of the \(m\)-th powers of their digits.
This is a finite sequence, the 89th and last term being:$$ 115132219018763992565095597973971522401$$They are named after Michael Frederick Armstrong (1941-2020), who used these numbers in his computing class at the University of Rochester in the mid 1960's. A brief biography of his is included at the end of this post.
D-POWERFUL NUMBERS
An integer $n$ is called digitally powerful or d-powerful if it can be expressed as a sum of positive powers of its digits. For example:$$3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21}$$These are far more numerous than narcissistic numbers. Here is a link to a text file that shows all the d-powerful numbers up to one million together with the configuration of their digits. It can be noted that every narcissistic number is a d-powerful number. The d-powerful numbers make up OEIS where they are referred to as "handsome" numbers.
A007532: Handsome numbers: sum of positive powers of its digits:$$ \text{a(}n) = \sum_{i=1}^k \text{d}[i]^{e[i]}$$where \( \text{d}[i]\) are the decimal digits of \( \text{a}(n), \text{e}[i] > 0\).
The initial d-powerful numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994
FRIEDMAN NUMBERS
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:$$25 = 5^2 \text{ and } 126 = 21 × 6$$Extensive information can be found about them via this link.
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 |
The Friedman numbers in the table that are in red are called "nice" because the digits on the right of the equations are in the same order as those on the right.
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 | 13243 = 41 × 323 | 13286 = 26 × (83 – 1) | 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 × ((7×9)2 – 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 × 8)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 = (4×8)3 + 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 – 7) | 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 – 4×9) | 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) / 3 | 39363 = 39 / 3 × 6 – 3 (MR) | 39366 = 39 / (–3 + 6) × 6 (MR) |
| 39368 = 6 × (38 + 3 / 9) | 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)4 + 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 = 6 × 9 × 25+5 |
| 56295 = 9 × 6255 | 56628 = (5 + 8) × 662 | 56732 = 26 × (37 – 5) | 56875 = 65 × 875 | 57288 = 7 × (25+8 – 8) |
| 57644 = 4 × (6 × 74 + 5) | 57645 = 57 – 5 × 46 | 58921 = 95 – 28–1 | 58957 = 95 – 5 – 87 | 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 + 5) + 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 = 4 × ((7 – 2)6 – 6) |
| 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 = (7 – 1 – 1)7 – 8 | 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 × 35 + 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 |
(TG) = Trevor Green, (MR) = Mike Reid, (PF) = Phillipe Fondanaiche, (JD) = Joe DeVincentis
There is a ton of information on the link provided so I'll leave there. A brief biography of
MICHAEL FREDERICK ARMSTRONG
Michael Frederick ArmstrongMichael Frederick Armstrong (1941–2020) was a senior computer systems programmer, educator, and prominent software industry leader. While his name is immortalized in recreational mathematics and computer science curricula worldwide through Armstrong numbers, his career extended far beyond that classroom exercise into the foundations of large-scale systems computing.
1. Role at the University of Rochester and the Birth of Armstrong Numbers
In the mid-1960s (circa 1966), Armstrong worked as a senior systems programmer at the University of Rochester Computing Center and occasionally taught programming courses. During this period, mainframe computing was expanding rapidly, and instructors needed novel, computationally intensive exercises to teach students how to write iterative loops, manipulate variables, and handle integer arithmetic using FORTRAN.
To give his students a compelling problem to solve on the university's computers, Armstrong devised a set of integer digit-manipulation definitions. He tasked his students with writing FORTRAN algorithms to search for numbers satisfying these conditions.
Because early 1960s computing hardware was relatively slow and constrained, searching for these digit invariants served as a practical benchmark for algorithm efficiency and loop optimization—a role the problem still fulfills in computer science education today.
2. Armstrong's Original Classification of Numbers
Although broader mathematical literature frequently refers to these figures as narcissistic numbers, pluperfect digital invariants (PPDIs), or plus-perfect numbers, Armstrong himself established four distinct tiers or "kinds" for his classroom assignments:
Armstrong Numbers of the First Kind (AN1): This is the definition most commonly referred to simply as an "Armstrong number" today. An integer \(N\) is equal to the sum of its own digits, with each digit raised to the power of the total number of digits \(m\) in the integer:$$\sum_{i=1}^{m} d_i^m = N$$Examples are:
\(153\) since \(1^3 + 5^3 + 3^3 = 153\)
\(1634\) since \(1^4 + 6^4 + 3^4 + 4^4 = 1634\).
In sequence databases like the OEIS, this forms sequence A005188.
Armstrong Numbers of the Second Kind (AN2):Armstrong Numbers of the Second Kind (AN2): Invariants involving sums of products or positional weight calculations based on the base-representation digits.
Armstrong Numbers of the Third Kind (AN3): Numbers where each digit \(d\) is raised to the power of itself \(d^d\) and summed to equal the original integer. In modern recreational mathematics, these are formally known as Perfect Digit-to-Digit Invariants (PDDIs) or Munchausen numbers. An examples is \(3435\) since \(3^3 + 4^4 + 3^3 + 5^5 = 3435\)
Armstrong Numbers of the Fourth Kind (AN4): Numbers equal to the sum of their digits raised to a fixed arbitrary power \(k\), regardless of the total number of digits the integer contains (now more broadly called Perfect Digital Invariants or PDIs of order \(k\).
Although Armstrong noted later in life that he was not actively drawing from previous 20th-century mathematical literature when drafting his FORTRAN homework sheets, his classroom problem quickly escaped the university setting and became a standard fixture in coding textbooks and programming contests.
3. Leadership in the Computing Industry (SHARE)
Beyond his mathematical eponym, Armstrong spent over five decades as a central figure in the mainframe systems community:
Early Involvement: He first attended a SHARE meeting in 1966 representing the University of Rochester. SHARE (founded in 1955) is one of the world's oldest volunteer-run enterprise computing user groups, originally formed by IBM mainframe users to share source code and technical insights.
Technical Contributions: Throughout the 1960s and 1970s, Armstrong contributed heavily to foundational enterprise operating system projects, including FORTRAN compiler optimization, Assembly language programming, HASP, OS/MVT-MFT, and VM virtual machine architectures.
SHARE Presidency: Entering the leadership track in the early 1980s as Deputy Project Manager for Computer Management and Evaluation, he was elected to the SHARE Board of Directors in 1982. He served a two-year term as President of SHARE from 1986 to 1988, guiding the enterprise computing community through a period of massive industry transition.
4. Enduring Legacy
While the largest base-10 Armstrong number of the first kind is a 39-digit integer 115132219018763992565095597973522401, the concept remains one of the most widely implemented beginner algorithms in computer science. Every year, thousands of students learning C, Python, Java, and FORTRAN encounter Michael F. Armstrong's 1966 classroom exercise as their introduction to modulus digit extraction and loop control.
Erich Friedman: Mathematician and Puzzler
Erich J. Friedman (born 1965 in West Lafayette, Indiana) is an American mathematician, educator, and prolific puzzle designer. While he is most widely recognized in recreational mathematics for defining Friedman numbers, his career encompasses a broad range of mathematical and computational disciplines. He spent 26 years as a professor of mathematics at Stetson University in DeLand, Florida, before retiring in 2018.
Academic Background and Research
Friedman earned his B.S. from Rose-Hulman Institute of Technology in 1987. He went on to complete his Ph.D. at Cornell University in 1991 under the supervision of Rick Durrett, where his initial focus was on interacting particle systems.
While his early academic roots were in probability, his published research frequently tackled discrete mathematics and geometry:
Geometrical Packing: Friedman made significant contributions to the problem of packing equal squares into a larger square. This is a computationally intense problem where he improved the best-known upper bounds for various configurations (for example, finding tighter bounds for \( n = 26 \) up through \( n = 88 \) squares).
Computational Complexity of Games:Computational Complexity of Games: Bridging logic and computer science, Friedman published multiple papers proving that various solitaire and logic puzzles—such as Spiral Galaxies, Push-k (block pushing), and Cubic—are NP-complete.
Contributions to Recreational Mathematics
Friedman is a central figure in modern recreational number theory. For years, he maintained popular online resources, including the "Math Magic" monthly problem series and the "What's Special About This Number?" database.
Friedman Numbers
Introduced by Friedman in August 2000, a Friedman number is a positive integer that can be represented in a non-trivial way using all of its own digits in combination with the basic arithmetic operators (( +, -, \times, \div )), additive inverses, parentheses, and exponentiation.
Examples of base-10 Friedman numbers include:
These numbers form sequence A036057 in the On-Line Encyclopedia of Integer Sequences (OEIS).A036057 In 2002, Friedman's deep involvement in integer sequence research led to him serving as an Associate Editor for the OEIS.
Nice Friedman Numbers
A highly sought-after subset of these are "nice" or "strong" Friedman numbers, where the mathematical expression uses the digits in the exact same order they appear in the integer itself. For instance:
Analyzing the density of these numbers, generating sequences of repdigit Friedman numbers (such as ( 99999999 )), and finding solutions in bases other than 10 are highly amenable to algorithm scripting and brute-force computational searches.
Puzzle Design and Personal Interests
Beyond formal mathematics, Friedman is an internationally acclaimed logic puzzle designer. He has created spatial, mathematical, and logic puzzles for the U.S. Puzzle Championship, the World Puzzle Championship, and magazines like Games and World of Puzzles.GamesWorld of Puzzles Several of his physical wooden puzzle designs have even been mass-produced.
Outside of his mathematical publications, Friedman has noted his passion for playing strategic games like chess, Go, and backgammon. He is also a self-described juggler and builder of card houses.
