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

Author

David W. Wilson

Keywords

• nonn
• base
• nice

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

Examples

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

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

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).

Programs

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 *)

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)={nn|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

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