Definition:Pluperfect Digital Invariant

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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

\(\displaystyle 153\) \(=\) \(\displaystyle 1 + 125 + 27\)
\(\displaystyle \) \(=\) \(\displaystyle 1^3 + 5^3 + 3^3\)


\(\displaystyle 370\) \(=\) \(\displaystyle 27 + 343 + 0\)
\(\displaystyle \) \(=\) \(\displaystyle 3^3 + 7^3 + 0^3\)


\(\displaystyle 371\) \(=\) \(\displaystyle 27 + 343 + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 3^3 + 7^3 + 1^3\)


\(\displaystyle 407\) \(=\) \(\displaystyle 64 + 0 + 343\)
\(\displaystyle \) \(=\) \(\displaystyle 4^3 + 0^3 + 7^3\)


$4$ Digit Pluperfect Digital Invariants

\(\displaystyle 1634\) \(=\) \(\displaystyle 1 + 1296 + 81 + 256\)
\(\displaystyle \) \(=\) \(\displaystyle 1^4 + 6^4 + 3^4 + 4^4\)


\(\displaystyle 8208\) \(=\) \(\displaystyle 4096 + 16 + 0 + 4096\)
\(\displaystyle \) \(=\) \(\displaystyle 8^4 + 2^4 + 0^4 + 8^4\)


\(\displaystyle 9474\) \(=\) \(\displaystyle 6561 + 256 + 2401 + 256\)
\(\displaystyle \) \(=\) \(\displaystyle 9^4 + 4^4 + 7^4 + 4^4\)


$5$ Digit Pluperfect Digital Invariants

\(\displaystyle 54 \, 748\) \(=\) \(\displaystyle 3125 + 1024 + 16 \, 807 + 1024 + 32 \, 768\)
\(\displaystyle \) \(=\) \(\displaystyle 5^5 + 4^5 + 7^5 + 4^5 + 8^5\)


\(\displaystyle 92 \, 727\) \(=\) \(\displaystyle 59 \, 049 + 32 + 16 \, 807 + 32 + 16 \, 807\)
\(\displaystyle \) \(=\) \(\displaystyle 9^5 + 2^5 + 7^5 + 2^5 + 7^5\)


\(\displaystyle 93 \, 084\) \(=\) \(\displaystyle 59 \, 049 + 243 + 0 + 32 \, 768 + 1024\)
\(\displaystyle \) \(=\) \(\displaystyle 9^5 + 3^5 + 0^5 + 8^5 + 4^5\)


$6$ Digit Pluperfect Digital Invariants

\(\displaystyle 548 \, 834\) \(=\) \(\displaystyle 15 \, 625 + 4096 + 262 \, 144 + 262 \, 144 + 729 + 4096\)
\(\displaystyle \) \(=\) \(\displaystyle 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6\)


$7$ Digit Pluperfect Digital Invariants

\(\displaystyle 1 \, 741 \, 725\) \(=\) \(\displaystyle 1 + 823 \, 543 + 16 \, 384 + 1 + 823 \, 543 + 128 + 78 \, 125\)
\(\displaystyle \) \(=\) \(\displaystyle 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7\)


\(\displaystyle 4 \, 210 \, 818\) \(=\) \(\displaystyle 16 \, 384 + 128 + 1 + 0 + 2 \, 097 \, 152 + 1 + 2 \, 097 \, 152\)
\(\displaystyle \) \(=\) \(\displaystyle 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7\)


\(\displaystyle 9 \, 800 \, 817\) \(=\) \(\displaystyle 4 \, 782 \, 969 + 2 \, 097 \, 152 + 0 + 0 + 2 \, 097 \, 152 + 1 + 823 \, 543\)
\(\displaystyle \) \(=\) \(\displaystyle 9^7 + 8^7 + 0^7 + 0^7 + 8^7 + 1^7 + 7^7\)


\(\displaystyle 9 \, 926 \, 315\) \(=\) \(\displaystyle 4 \, 782 \, 969 + 4 \, 782 \, 969 + 128 + 279 \, 936 + 2187 + 1 + 78 \, 125\)
\(\displaystyle \) \(=\) \(\displaystyle 9^7 + 9^7 + 2^7 + 6^7 + 3^7 + 1^7 + 5^7\)


$8$ Digit Pluperfect Digital Invariants

\(\displaystyle 24 \, 678 \, 050\) \(=\) \(\displaystyle 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 0\)
\(\displaystyle \) \(=\) \(\displaystyle 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 0^8\)


\(\displaystyle 24 \, 678 \, 051\) \(=\) \(\displaystyle 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 1\)
\(\displaystyle \) \(=\) \(\displaystyle 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 1^8\)


\(\displaystyle 88 \, 593 \, 477\) \(=\) \(\displaystyle 16 \, 777 \, 216 + 16 \, 777 \, 216 + 390 \, 625 + 43 \, 046 \, 721 + 6561 + 65 \, 536 + 5 \, 764 \, 801 + 5 \, 764 \, 801\)
\(\displaystyle \) \(=\) \(\displaystyle 8^8 + 8^8 + 5^8 + 9^8 + 3^8 + 4^8 + 7^8 + 7^8\)


$9$ Digit Pluperfect Digital Invariants

\(\displaystyle 146 \, 511 \, 208\) \(=\) \(\displaystyle 1 + 262 \, 144 + 10 \, 077 \, 696 + 1 \, 953 \, 125 + 1 + 1 + 512 + 0 + 134 \, 217 \, 728\)
\(\displaystyle \) \(=\) \(\displaystyle 1^9 + 4^9 + 6^9 + 5^9 + 1^9 + 1^9 + 2^9 + 0^9 + 8^9\)


\(\displaystyle 472 \, 335 \, 975\) \(=\) \(\displaystyle 262 \, 144 + 40 \, 353 \, 607 + 512 + 19 \, 683 + 19 \, 683 + 1 \, 953 \, 125 + 387 \, 420 \, 489 + 40 \, 353 \, 607 + 1 \, 953 \, 125\)
\(\displaystyle \) \(=\) \(\displaystyle 4^9 + 7^9 + 2^9 + 3^9 + 3^9 + 5^9 + 9^9 + 7^9 + 5^9\)


\(\displaystyle 534 \, 494 \, 836\) \(=\) \(\displaystyle 1 \, 953 \, 125 + 19 \, 683 + 262 \, 144 + 262 \, 144 + 387 \, 420 \, 489 + 262 \, 144 + 134 \, 217 \, 728 + 19 \, 683 + 10 \, 077 \, 696\)
\(\displaystyle \) \(=\) \(\displaystyle 5^9 + 3^9 + 4^9 + 4^9 + 9^9 + 4^9 + 8^9 + 3^9 + 6^9\)


\(\displaystyle 912 \, 985 \, 153\) \(=\) \(\displaystyle 387 \, 420 \, 489 + 1 + 512 + 387 \, 420 \, 489 + 134 \, 217 \, 728 + 1 \, 953 \, 125 + 1 + 1 \, 953 \, 125 + 19 \, 683\)
\(\displaystyle \) \(=\) \(\displaystyle 9^9 + 1^9 + 2^9 + 9^9 + 8^9 + 5^9 + 1^9 + 5^9 + 3^9\)


$10$ Digit Pluperfect Digital Invariants

\(\displaystyle 4 \, 679 \, 307 \, 774\) \(=\) \(\displaystyle 1 \, 048 \, 576 + 60 \, 466 \, 176 + 282 \, 475 \, 249 + 3 \, 486 \, 784 \, 401 + 59 \, 049 + 0 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 1 \, 048 \, 576\)
\(\displaystyle \) \(=\) \(\displaystyle 4^{10} + 6^{10} + 7^{10} + 9^{10} + 3^{10} + 0^{10} + 7^{10} + 7^{10} + 7^{10} + 4^{10}\)


Also known as

A pluperfect digital invariant is also known as:

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

It is not known who the Armstrong is whose eponym this entry is, although there are rumours that he may have been a Michael Armstrong of Polk City in Florida.


Also see


Sources