Definition:Pluperfect Digital Invariant

Definition

Let $n \in \Z_{>0}$ be a positive integer.

$n$ is a pluperfect digital invariant if and only if when expressed in decimal notation the $n$th powers of its digits add up to $n$.

Sequence

The complete sequence of pluperfect digital invariants is:

$1, 2, 3, 4, 5, 6, 7, 8, 9,$

$153, 370, 371, 407,$

$1634, 8208, 9474,$

$54748, 92727, 93084,$

$548834, 1741725,$

$4210818, 9800817, 9926315,$

$24678050, 24678051, 88593477,$

$146511208, 472335975, 534494836, 912985153,$

$4679307774,$

$32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914,$

$28116440335967,$

$4338281769391370, 4338281769391371,$

$21897142587612075, 35641594208964132, 35875699062250035,$

$1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826,$

$63105425988599693916,$

$128468643043731391252, 449177399146038697307,$

$21887696841122916288858, 27879694893054074471405, 27907865009977052567814,$

$28361281321319229463398, 35452590104031691935943,$

$174088005938065293023722, 188451485447897896036875, 239313664430041569350093,$

$1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380,$

$3706907995955475988644381, 4422095118095899619457938,$

$121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787,$

$174650464499531377631639254, 177265453171792792366489765,$

$14607640612971980372614873089, 19008174136254279995012734740,$

$19008174136254279995012734741, 23866716435523975980390369295,$

$1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915,$

$17333509997782249308725103962772,$

$186709961001538790100634132976990, 186709961001538790100634132976991,$

$1122763285329372541592822900204593,$

$12639369517103790328947807201478392, 12679937780272278566303885594196922,$

$1219167219625434121569735803609966019,$

$12815792078366059955099770545296129367,$

$115132219018763992565095597973971522400, 115132219018763992565095597973971522401$

Examples of Pluperfect Digital Invariants

$3$ Digit Pluperfect Digital Invariants

\(\ds 153\)

\(=\)

\(\ds 1 + 125 + 27\)

\(\ds \)

\(=\)

\(\ds 1^3 + 5^3 + 3^3\)

\(\ds 370\)

\(=\)

\(\ds 27 + 343 + 0\)

\(\ds \)

\(=\)

\(\ds 3^3 + 7^3 + 0^3\)

\(\ds 371\)

\(=\)

\(\ds 27 + 343 + 1\)

\(\ds \)

\(=\)

\(\ds 3^3 + 7^3 + 1^3\)

\(\ds 407\)

\(=\)

\(\ds 64 + 0 + 343\)

\(\ds \)

\(=\)

\(\ds 4^3 + 0^3 + 7^3\)

$4$ Digit Pluperfect Digital Invariants

\(\ds 1634\)

\(=\)

\(\ds 1 + 1296 + 81 + 256\)

\(\ds \)

\(=\)

\(\ds 1^4 + 6^4 + 3^4 + 4^4\)

\(\ds 8208\)

\(=\)

\(\ds 4096 + 16 + 0 + 4096\)

\(\ds \)

\(=\)

\(\ds 8^4 + 2^4 + 0^4 + 8^4\)

\(\ds 9474\)

\(=\)

\(\ds 6561 + 256 + 2401 + 256\)

\(\ds \)

\(=\)

\(\ds 9^4 + 4^4 + 7^4 + 4^4\)

$5$ Digit Pluperfect Digital Invariants

\(\ds 54 \, 748\)

\(=\)

\(\ds 3125 + 1024 + 16 \, 807 + 1024 + 32 \, 768\)

\(\ds \)

\(=\)

\(\ds 5^5 + 4^5 + 7^5 + 4^5 + 8^5\)

\(\ds 92 \, 727\)

\(=\)

\(\ds 59 \, 049 + 32 + 16 \, 807 + 32 + 16 \, 807\)

\(\ds \)

\(=\)

\(\ds 9^5 + 2^5 + 7^5 + 2^5 + 7^5\)

\(\ds 93 \, 084\)

\(=\)

\(\ds 59 \, 049 + 243 + 0 + 32 \, 768 + 1024\)

\(\ds \)

\(=\)

\(\ds 9^5 + 3^5 + 0^5 + 8^5 + 4^5\)

$6$ Digit Pluperfect Digital Invariants

\(\ds 548 \, 834\)

\(=\)

\(\ds 15 \, 625 + 4096 + 262 \, 144 + 262 \, 144 + 729 + 4096\)

\(\ds \)

\(=\)

\(\ds 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6\)

$7$ Digit Pluperfect Digital Invariants

\(\ds 1 \, 741 \, 725\)

\(=\)

\(\ds 1 + 823 \, 543 + 16 \, 384 + 1 + 823 \, 543 + 128 + 78 \, 125\)

\(\ds \)

\(=\)

\(\ds 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7\)

\(\ds 4 \, 210 \, 818\)

\(=\)

\(\ds 16 \, 384 + 128 + 1 + 0 + 2 \, 097 \, 152 + 1 + 2 \, 097 \, 152\)

\(\ds \)

\(=\)

\(\ds 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7\)

\(\ds 9 \, 800 \, 817\)

\(=\)

\(\ds 4 \, 782 \, 969 + 2 \, 097 \, 152 + 0 + 0 + 2 \, 097 \, 152 + 1 + 823 \, 543\)

\(\ds \)

\(=\)

\(\ds 9^7 + 8^7 + 0^7 + 0^7 + 8^7 + 1^7 + 7^7\)

\(\ds 9 \, 926 \, 315\)

\(=\)

\(\ds 4 \, 782 \, 969 + 4 \, 782 \, 969 + 128 + 279 \, 936 + 2187 + 1 + 78 \, 125\)

\(\ds \)

\(=\)

\(\ds 9^7 + 9^7 + 2^7 + 6^7 + 3^7 + 1^7 + 5^7\)

$8$ Digit Pluperfect Digital Invariants

\(\ds 24 \, 678 \, 050\)

\(=\)

\(\ds 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 0\)

\(\ds \)

\(=\)

\(\ds 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 0^8\)

\(\ds 24 \, 678 \, 051\)

\(=\)

\(\ds 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 1\)

\(\ds \)

\(=\)

\(\ds 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 1^8\)

\(\ds 88 \, 593 \, 477\)

\(=\)

\(\ds 16 \, 777 \, 216 + 16 \, 777 \, 216 + 390 \, 625 + 43 \, 046 \, 721 + 6561 + 65 \, 536 + 5 \, 764 \, 801 + 5 \, 764 \, 801\)

\(\ds \)

\(=\)

\(\ds 8^8 + 8^8 + 5^8 + 9^8 + 3^8 + 4^8 + 7^8 + 7^8\)

$9$ Digit Pluperfect Digital Invariants

\(\ds 146 \, 511 \, 208\)

\(=\)

\(\ds 1 + 262 \, 144 + 10 \, 077 \, 696 + 1 \, 953 \, 125 + 1 + 1 + 512 + 0 + 134 \, 217 \, 728\)

\(\ds \)

\(=\)

\(\ds 1^9 + 4^9 + 6^9 + 5^9 + 1^9 + 1^9 + 2^9 + 0^9 + 8^9\)

\(\ds 472 \, 335 \, 975\)

\(=\)

\(\ds 262 \, 144 + 40 \, 353 \, 607 + 512 + 19 \, 683 + 19 \, 683 + 1 \, 953 \, 125 + 387 \, 420 \, 489 + 40 \, 353 \, 607 + 1 \, 953 \, 125\)

\(\ds \)

\(=\)

\(\ds 4^9 + 7^9 + 2^9 + 3^9 + 3^9 + 5^9 + 9^9 + 7^9 + 5^9\)

\(\ds 534 \, 494 \, 836\)

\(=\)

\(\ds 1 \, 953 \, 125 + 19 \, 683 + 262 \, 144 + 262 \, 144 + 387 \, 420 \, 489 + 262 \, 144 + 134 \, 217 \, 728 + 19 \, 683 + 10 \, 077 \, 696\)

\(\ds \)

\(=\)

\(\ds 5^9 + 3^9 + 4^9 + 4^9 + 9^9 + 4^9 + 8^9 + 3^9 + 6^9\)

\(\ds 912 \, 985 \, 153\)

\(=\)

\(\ds 387 \, 420 \, 489 + 1 + 512 + 387 \, 420 \, 489 + 134 \, 217 \, 728 + 1 \, 953 \, 125 + 1 + 1 \, 953 \, 125 + 19 \, 683\)

\(\ds \)

\(=\)

\(\ds 9^9 + 1^9 + 2^9 + 9^9 + 8^9 + 5^9 + 1^9 + 5^9 + 3^9\)

$10$ Digit Pluperfect Digital Invariants

\(\ds 4 \, 679 \, 307 \, 774\)

\(=\)

\(\ds 1 \, 048 \, 576 + 60 \, 466 \, 176 + 282 \, 475 \, 249 + 3 \, 486 \, 784 \, 401 + 59 \, 049 + 0 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 1 \, 048 \, 576\)

\(\ds \)

\(=\)

\(\ds 4^{10} + 6^{10} + 7^{10} + 9^{10} + 3^{10} + 0^{10} + 7^{10} + 7^{10} + 7^{10} + 4^{10}\)

Also known as

A pluperfect digital invariant is also known as:

• a plus perfect number
• a narcissistic number
• an Armstrong number.

Also see

• Definition:Recurring Digital Invariant

• Results about pluperfect digital invariants can be found here.

Historical Note

The concept of a pluperfect digital invariant was dismissed thus by Godfrey Harold Hardy in his A Mathematician's Apology of $1940$:

There are just four numbers, after unity, which are the sums of the cubes of their digits: $153 = 1^3 + 5^3 + 3^3$, $370 = 3^3 + 7^3 + 0^3$, $371 = 3^3 + 7^3 + 1^3$, and $407 = 4^3 + 0^3 + 7^3$. These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.

Previous to this, they surfaced in Henry Ernest Dudeney's Modern Puzzles of $1926$ as puzzle $94$, although he failed to find $371$.

The eponym of Armstrong number is Michael Frederick Armstrong of Polk City in Florida, a teacher of computing at Rochester University in the $1960$s.

He set as an assignment an exercise to hunt for them.

While the concept did not originate with him, his name can often be seen associated with them.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $61$

• Weisstein, Eric W. "Narcissistic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NarcissisticNumber.html