A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

Offset

1,3

Comments

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.

Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.

Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.

Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

If a term x is a multiple of 10, then x+1 is a term too. - Paolo P. Lava, Apr 07 2016

Links

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)

Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.

Hans Havermann, Extended table of values for A023052 and A046074

Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles

J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.

J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Narcissistic Number

Index entries for sequences related to powerful numbers

Example

153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

Maple

P:=proc(q) local a, b, c, k, n, ok; for n from 1 to q do a:=[]; b:=n; ok:=0;
for k from 1 to ilog10(n)+1 do if (b mod 10)>1 then ok:=1; fi;
a:=[(b mod 10), op(a)]; b:=trunc(b/10); od; b:=0; c:=0;
if ok=1 then while c<n do b:=b+1; c:=add(a[k]^b, k=1..nops(a)); od;  if
c=n then print(n); fi; fi; od; end: P(10^9); # Paolo P. Lava, Feb 08 2016

Mathematica

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

Prog

(PARI) is(n)=if(n<10, return(1)); my(d=digits(n), m=vecmax(d)); if(m<2, return(0)); for(k=3, logint(n, m), if(sum(i=1, #d, d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017
(PARI) select( is_A023052(n, b=10)={n<b|| forstep(p=logint(n, max(vecmax(b=digits(n, b)), 2)), 2, -1, my(t=vecsum([d^p|d<-b])); t>n|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

Crossrefs

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).

Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.

Cf. A005188 (here the power must be equal to the number of digits).

In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

Sequence in context: A032561 A342157 A306360 * A005188 A032569 A343036

Adjacent sequences: A023049 A023050 A023051 * A023053 A023054 A023055

Keyword

nonn,base,nice

Author

David W. Wilson

Extensions

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)

Computed to 10^74 by Xiaoqing Tang

A-number typo corrected by R. J. Mathar, Jun 22 2009

Computed to 10^105 by Joseph Myers

Cross-references edited by Joseph Myers, Jun 28 2009

Edited by M. F. Hasler, Nov 21 2019

Status

approved