Narcissistic, D-Powerfull and Friedman Numbers

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:

• \( 25 = 5^2 \)

• \( 126 = 6 \times 21 \)

• \( 347 = 7^3 + 4 \)

• \( 1024 = (4 - 2)^{10} \)

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:

• \( 127 = -1 + 2^7 \)

• \( 2592 = 2^5 \times 9^2 \)

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.

SOD ET AL

SOD stands in a mathematical context for Sum of Digits and it can also be written as SoD or sod. Unlike a number's primeness or non-primeness, a number's SoD is peculiar to the number system being used and is thus of interest mainly in recreational mathematics. My diurnal age today is 26384 with a sum of digits of 23. When these two numbers are added together, the result is a prime number, 26407. This qualifies 26384 for inclusion in OEIS A047791:

A047791

Numbers \(n\) such that \(n\) plus digit sum of \(n\) (A007953) equals a prime.

There are 2919 such numbers in the range of numbers from 1 to 26384, constituting about 11.1% of the total number. In the same range, there are 2897 primes and so the totals are nearly identical. This is not surprising because the operation of adding the sum of digits of a number to itself simply changes the number into another number of slightly higher value, without regard to its being prime or composite.

ODDS AND EVENS

Recently, I made a series of posts that involved adding the odd digits to a number and subtracting the even digits. These posts were titled:

• Odds and Evens
• Attractors, Vortices and Captives
• Binary Odds and Evens
• Odds and Evens: Statistics

I covered a lot of material in those posts so refer to those for more details.

SELF AND JUNCTION NUMBERS

In this post, I want to collect together some of the other mathematical activities that involve the sum of the digits of a number or the manipulation of the digits is some way. Let's start with the concept of a self number. If there does NOT exist a number \(x\) such that \(x\) + sod(\(x\)) = \(n\) for some number \(n\), then \(n\) is said to be a self number. There are 10 self numbers in the range from 26300 to 26400:

26307, 26318, 26320, 26331, 26342, 26353, 26364, 26375, 26386, 26397

Looking at the above numbers, it can be seen that 26308 is not in the list. This is because:

26285 + sod(26285) = 26285 + 23 = 26308

Apart from self numbers, most numbers are like 26385. There is only one value of \(x\) for which \(x\) + sod(\(x\)) = \(n\). If there is more than one value of x then the number is said to be a junction number. In the range from 26300 to 26400, there are nine junction numbers:

26311 is a junction number [26291, 26300]

26313 is a junction number [26292, 26301]

26315 is a junction number [26293, 26302]

26317 is a junction number [26294, 26303]

26319 is a junction number [26295, 26304]

26321 is a junction number [26296, 26305]

26323 is a junction number [26297, 26306]

26325 is a junction number [26298, 26307]

26327 is a junction number [26299, 26308]

Thus we see, using the first number 26311 as an example, that: 

• 26291 + sod(26291) = 26291 + 20 = 26311 and 
• 26300 + sod(26300) = 26300 + 11 = 26311

I've written about self and junction numbers in an eponymous post from October 25th 2018. The numbers above give an idea of the relative proportions of such numbers: about 10% are self numbers, 10% are junction numbers and 80% are neither.

HAPPY NUMBERS

Happy numbers don't involve the sum of the digits per se but instead are concerned with the sum of the digits squared. If this process is applied recursively and the end result is 1, then the number is said to be happy. For example, 94 is an example of such a number because:

 94 → 97 → 130 → 10 → 1 

Only about 15% of numbers are happy. The rest end up in a loop (4, 16, 37, 58, 89, 145, 42, 20, 4) and 61 is an example of such a number because:

61 → 37 → 58 → 89 and the loop has been entered

I've written about happy numbers in a series of posts:

• Happy Numbers
• Asymptotic Density of Happy Go Lucky Numbers

SELFIE NUMBERS

I wrote about these sorts of numbers in an eponymous post on March 27th 2020. In it, I quoted the following:

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, squareroot, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have written selfie numbers by use of concatenation, along with factorial and square-root. The concatenation idea is used in a very simple way. The work is limited up to 5 digits. Work on higher digits shall be dealt elsewhere. Source.

I use the example of 25926 that can be expressed as: 

((−2+5)!)!×C(9,2)+6 = (3!)! x 36 + 6 = 6! x 36 + 6 = 720 x 36 + 6 = 25926

Another example is \(39304:=((4||03)−9)^3\) where || stands for concatenation.

This is a big topic and it has been covered in detail in my prementioned blog post.

FRIEDMAN NUMBERS

These could be considered a subset of the selfie numbers but they form a category in their own right and are constructed much more simply. I wrote about these in a blog post from October 8th 2020 titled Forming Equations from Integers. To quote:

Consider \(28547=(8+5)^4−(7×2)\) expressed in base 10, both sides use the same digits. An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals. When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example, \(3125=(3+[1×2])^5\).

NARCISSISTIC NUMBERS 

Selfie numbers are sometimes called wild narcissistic numbers but the proper narcissistic numbers. Numbers Aplenty defines them thus:

A number \(n\)  of \(k\)  digits is called narcissistic if it is equal to the sum of the \(k^{th}\) powers of its digits. For example, \(153\)  is narcissistic because \(153 = 1^3+5^3+3^3\). Narcissistic numbers are also called Armstrong or plus-perfect numbers. It has verified that there in fact only 88 such numbers. Those up to one million are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315.

D-POWERFUL NUMBERS 

D-powerful numbers are akin to narcissistic numbers but with more flexibility regarding the powers to which the digits may be raised. To quote from Numbers Aplenty:

An integer \(n\) is called digitally powerful (here d-powerful) if it can be expressed as a sum of positive powers of its digits. For example:$$3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21}$$The first 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

HARSHAD AND MORAN NUMBERS 

I've written about Harshad Numbers in the following posts:

• Harshad Numbers
• Harshad Numbers Revisited

In the first of the two posts, I posted from Wikipedia:

In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. 

They are quite frequent and account for about 12% of all the numbers up to 100,000. Here are the first few:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204

If the dividend happens to be a prime number then the number is said to be a Moran number. To quote from Numbers Aplenty again:

A number \(n\) is a Moran number if \(n\) divided by the sum of its digits gives a prime number. For example, 111 is a Moran number because 111/(1+1+1) = 37 and 37 is a prime number. Moran numbers are a subset of Harshad numbers. 

The first few Moran numbers are: 

18, 21, 27, 42, 45, 63, 84, 111, 114, 117, 133, 152, 153, 156, 171, 190, 195, 198, 201, 207, 209, 222, 228. 

MAGNANIMOUS NUMBERS

I've written about these in an eponymous post December 27th 2020. To quote from Numbers Aplenty, a magnanimous number can be define as:

A number (which we assume of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.

For example, 4001 is magnanimous because the numbers 4+001=5, 40+01=41 and 400+1=401 are all prime numbers.

Since all the prime numbers are odd, except for 2, all the magnanimous numbers, except for 11, are either a sequence of odd digits followed by an even digit, or a sequence of even digits followed by an odd digits.

It is conjectured that the magnanimous numbers are finite and that probably the largest one is 97393713331910, while the largest one which is also a prime number itself is probably 608844043.

The first such numbers are:

11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203 

DIGITAL ROOT 

While I've not made a specific post about digital roots, I've nonetheless mentioned them in the following posts:

• Heptagonal Numbers
• Star Numbers
• The Goldbach Conjecture and Lucky Numbers
• An Unhappy Family

To quote from Wikipedia:

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0).

Associated with the digital root is the concept of additive persistence defined as:

The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9. 

SMITH AND HOAX NUMBERS 

I mentioned Smith numbers in a blog post dating back to April 21st 2016 and titled Repunits and Smith Numbers. 

Smith numbers are composite numbers with the property that the sum of their digits equals the sum of digits of their prime factors e.g. 22 → 2 + 2 = 4 and 22 = 2 * 11 → 2 + 1 + 1 = 4.

Hoax numbers are similar except they only consider distinct prime factors. Thus, for example, the Smith numbers 4 and 27 are excluded because the sums of their distinct prime factors are 2 and 3 respectively whereas their sums of digits are 4 and 9. The set of hoax numbers is a subset of the set of Smith numbers. 

666 is a Smith number since 666 = 2 * 3 * 3 * 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7. 

The initial Smith numbers are:

 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438 

The initial hoax numbers are:

22, 58, 84, 85, 94, 136, 160, 166, 202, 234, 250, 265, 274, 308, 319, 336, 346, 355  

 

THE RATS SEQUENCE

I wrote about this in an eponymous post from September 26th 2020. Here is an excerpt:

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by

668+866=1534

so the next term is 1345.

Applied to 1, the sequence gives: 

1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).

Conway conjectured that an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (Guy 2004, p. 404).

The lengths of the cycles obtained by starting with n= 1, 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611), where a 0 indicates that the sequence diverges.