More On The RDIV Algorithm

Under the RDIV or Recurring Digital Invariant Variant (a weird name I know) Algorithm, all numbers that are not narcissistic will enter a loop or terminate in a narcissistic number. I was interested in the proportion of numbers that terminate in a narcissistic number and so I had Gemini create an algorithm (permalink) to determine this. In the range up to 40000, there are 12224 such numbers which account for 30.56% of the range. These include the few numbers that are narcissistic themselves namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,153, 370, 371, 407, 1634, 8208 and 9474 within the range. 

Here is a fuller list of narcissistic numbers (OEIS A005188):

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651, 40028394225, 42678290603

While 12224 numbers are far too numerous to list here, we can thin the numbers by considering only triplets - meaning groups of three consecutive numbers that all lead to narcissistic numbers. Take for example, the numbers 28220, 28221 and 28222. Let's look at  their trajectories under the RDIV algorithm (permalink for generation). It will be seen that all three terminate in narcissistic numbers.

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 28220
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Full Trajectory Visited:
28220, 32864, 41843, 35060, 11144, 2051, 642, 288, 
1032, 98, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 14)
Pre-period Length:  13 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 28221
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Full Trajectory Visited:
28221, 32865, 43944, 62364, 16851, 43671, 25851, 
39051, 62418, 41601, 8802, 8208

Loop Entry Point:   8208 (encountered at step 12)
Pre-period Length:  11 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    8208
==================================================

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 28222
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Full Trajectory Visited:
28222, 32896, 99868, 191410, 535540, 51700, 19933, 118585,
555540, 66596, 85502, 39050, 62417, 25640, 11957, 78983,
141635, 67108, 57352, 23332, 793, 1099, 13123, 520, 133, 
55, 50, 25, 29, 85, 89, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 35)
Pre-period Length:  34 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370
==================================================

In the range up to 40000 there are 1786 such triplets and if we restrict the range to those above 28000, there are only 220 triplets. The central members of each triplet are listed below:

28134, 28221, 28314, 28365, 28563, 28635, 28653, 29121, 29211, 29278, 29728, 29729, 29792, 29972, 30006, 30051, 30055, 30060, 30061, 30151, 30160, 30221, 30222, 30223, 30224, 30233, 30234, 30242, 30249, 30250, 30251, 30252, 30280, 30323, 30324, 30343, 30422, 30433, 30501, 30505, 30510, 30511, 30520, 30521, 30522, 30561, 30601, 30610, 30651, 30820, 31051, 31060, 31111, 31112, 31113, 31114, 31115, 31132, 31133, 31142, 31143, 31284, 31312, 31313, 31412, 31413, 31474, 31475, 31501, 31510, 31744, 31745, 31824, 31839, 32021, 32022, 32023, 32024, 32033, 32034, 32042, 32049, 32050, 32051, 32052, 32080, 32184, 32200, 32201, 32202, 32203, 32204, 32221, 32246, 32254, 32303, 32304, 32402, 32409, 32410, 32426, 32453, 32501, 32502, 32519, 32524, 32529, 32543, 32649, 32685, 32800, 32814, 32865, 33023, 33024, 33043, 33112, 33113, 33199, 33203, 33204, 33310, 33332, 33403, 33556, 33564, 33573, 33574, 33654, 33753, 33754, 34022, 34033, 34112, 34113, 34174, 34175, 34202, 34209, 34210, 34226, 34253, 34303, 34470, 34482, 34523, 34629, 34714, 34715, 34809, 34842, 35001, 35005, 35010, 35011, 35020, 35021, 35022, 35061, 35101, 35110, 35201, 35202, 35219, 35224, 35229, 35243, 35356, 35364, 35373, 35374, 35423, 35500, 35536, 35557, 35558, 35564, 35565, 35601, 35634, 35654, 35655, 35733, 35734, 36001, 36010, 36051, 36249, 36285, 36354, 36429, 36501, 36534, 36554, 36555, 36740, 36825, 37144, 37145, 37353, 37354, 37414, 37415, 37533, 37534, 37640, 37898, 37988, 38020, 38124, 38139, 38214, 38265, 38409, 38442, 38625, 38798, 38978, 39788, 39878

What if we look for quadruplets of such numbers, that is four consecutive numbers such that each of them leads to a narcissistic number under the RDIV algorithm. There are 843 such quadruplets. Let's consider 29727, 29728, 29729 and 29730 (permalink for generation):

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 29727
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Full Trajectory Visited:
29727, 92727

Loop Entry Point:   92727 (encountered at step 2)
Pre-period Length:  1 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    92727
==================================================

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 29728
==================================================
Full Trajectory Visited:
29728, 108688, 833089, 1057187, 3822365, 2459843, 
6993329, 14633345, 2220997, 10389865, 78677956, 
80868197, 100822787, 349143695, 787435454, 219637307, 
478245278, 351621107, 52404626, 3881441, 4229261, 
5079674, 6804500, 2471597, 6524693, 5439665, 5517662, 
1539794, 10486178, 41064387, 24359267, 50954372, 
49665125, 47252996, 93994532, 129609702, 825273306, 
186642546, 166928742, 582410385, 272624046, 61034823, 
18535747, 29160166, 48085827, 56552866, 22988195, 
120039012, 387441198, 696753525, 453809058, 660044022, 
20680704, 24287425, 23064482, 18594977, 114856422, 
146773863, 235381845, 272643729, 478488762, 493962894, 
1307101374, 566117175, 104768859, 708502518, 312695826, 
543767442, 93544662, 46934467, 52373923, 49222342, 
43185378, 39788517, 88527942, 82822627, 40999873, 
151754277, 125229729, 817149759, 991981191, 1683899688, 
11389527500, 42046156391, 32164049651

Loop Entry Point:   32164049651 (encountered at step 85)
Pre-period Length:  84 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    32164049651
==================================================

==================================================
RDIV TRAJECTORY ANALYSIS FOR INPUT: 29729
==================================================
Full Trajectory Visited:
29729, 134969, 1114364, 314894, 802507, 395482, 814099, 
1329123, 4787601, 4040559, 4971987, 13326561, 3763237, 
1933711, 5610889, 9335335, 4947967, 11525728, 23323781, 
22562213, 2077827, 4568037, 3297327, 6434685, 2770104, 
1663599, 10206123, 1686691, 7719931, 11215213, 397702, 
767532, 298372, 912091, 1062948, 7176570, 2828691, 
9257466, 6261021, 560130, 63011, 8021, 4113, 339, 
783, 882, 1032, 98, 145, 190, 730, 370

Loop Entry Point:   370 (encountered at step 52)
Pre-period Length:  51 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    370
==================================================

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 29730
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Full Trajectory Visited:
29730, 76131, 24828, 66624, 24384, 35091, 62418, 41601, 
8802, 8208

Loop Entry Point:   8208 (encountered at step 10)
Pre-period Length:  9 step(s) before entering cycle
Cycle Length:       1 distinct number(s) in the loop
Canonical Cycle:    8208
==================================================

Because there are four consecutive numbers, only the first and smallest will be listed and so in this case the number would be 29727. In the range between 28000 and 40000, here are the 60 initial or smallest members of each quadruplet:

29727, 30059, 30220, 30221, 30222, 30232, 30248, 30249, 30250, 30322, 30509, 30519, 30520, 31110, 31111, 31112, 31113, 31131, 31141, 31311, 31411, 31473, 31743, 32020, 32021, 32022, 32032, 32048, 32049, 32050, 32199, 32200, 32201, 32202, 32302, 32408, 32500, 33022, 33111, 33202, 33572, 33752, 34111, 34173, 34208, 34713, 35009, 35019, 35020, 35200, 35372, 35556, 35563, 35653, 35732, 36553, 37143, 37352, 37413, 37532

As for quintuplets, there are 451 of them in the range up to 40000. However, above 28000 there are only 16 and the initial or smallest members of each are:

30220, 30221, 30248, 30249, 30519, 31110, 31111, 31112, 32020, 32021, 32048, 32049, 32199, 32200, 32201, 35019

There are eight sextuplets: 30220, 30248, 31110, 31111, 32020, 32048, 32199, 32200.

There are two septuplets: 31110 and 32199 and no octuplets within the range.

Recurring Digital Invariant Variant (RDIV) Algorithm

Let's consider the following algorithm (formally called the Recurring Digital Invariant Variant or RDIV algorithm - see this link for an explanation of the name):

• choose a number \(n\)
• let \(k\) be the number of digits in \(n\)
• raise each digit of \(n\) to the \(k\)-th power and add the results
• call the new number \(n\) and repeat

Let's use \(n=14\) as an example:

• \(14 \rightarrow 1^2 + 4^2 = 17\)
• \(17 \rightarrow 1^2 + 7^2 = 50\)
• \(50 \rightarrow 5^2 + 0^2 = 25\)
• \(25 \rightarrow 2^2 + 5^2 = 29\)
• \(29 \rightarrow 2^2 + 9^2 = 85\)
• \(85 \rightarrow 8^2 + 5^2 = 89\)
• \(89 \rightarrow 8^2 + 9^2 = 145\)
• \(145 \rightarrow 1^3 + 4^3 + 5^3 = 190\)
• \(190 \rightarrow 1^3 + 9^3 + 0^3 = 730\)
• \(730 \rightarrow 7^3 + 3^3 + 0^3 = 370\)
• \(370 \rightarrow 3^3 + 7^3 + 0^3 = 370\) 

370 is a narcissistic number as explained in my post Narcissistic, D-Powerfull and Friedman Numbers. The trajectory of any number under this algorithm will either end with a narcissistic number (as was the case with 14) or it will enter a loop (as is the case with 28218). The latter has the following trajectory (permalink):

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RDIV TRAJECTORY ANALYSIS FOR INPUT: 28218

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Full Trajectory Visited:

28218, 65601, 18678, 90120, 59082, 94974, 136953, 595181, 824837, 646826, 406272, 168529, 855931, 825565, 355739, 681798, 1220035, 80569, 102718, 379859, 1459029, 9660576, 6524445, 485466, 379273, 768261, 473170, 240124, 8321, 4194, 7074, 5058, 5346, 2258, 4753, 3363, 1539, 7268, 7809, 13058, 36137, 25070, 19964, 126899, 1371747, 2489202, 6896889, 16417266, 10869443, 61641187, 25966788, 86116067, 27580867, 47154531, 6683686, 5316235, 440689, 848433, 533938, 811397, 911965, 1125165, 436317, 169860, 886898, 1626673, 1665667, 2021413, 18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069 -> [loops back to 886898]

Loop Entry Point:   886898 (encountered at step 65)

Pre-period Length:  64 step(s) before entering cycle

Cycle Length:       14 distinct number(s) in the loop

Canonical Cycle:    18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413

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Figure 1 shows a graph of its trajectory:

Figure 1: permalink

I've incorporated this algorithm into my daily number analysis.

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.