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 A023052 Powerful numbers (3): numbers n that are the sum of some fixed power of their digits. 31

%I

%S 0,1,2,3,4,5,6,7,8,9,153,370,371,407,1634,4150,4151,8208,9474,54748,

%T 92727,93084,194979,548834,1741725,4210818,9800817,9926315,14459929,

%U 24678050,24678051,88593477,146511208,472335975,534494836,912985153

%N Powerful numbers (3): numbers n that are the sum of some fixed power of their digits.

%C Let n = d_1 d_2 ... d_k in base 10; then n = Sum_{i=1..k} d_i^s for some s.

%C Superset of A005188, 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]

%C If a term x is a multiple of 10, then x+1 is a term too. - _Paolo P. Lava_, Apr 07 2016

%H Joseph Myers, <a href="/A023052/b023052.txt">Table of n, a(n) for n = 1..255</a>

%H Hans Havermann, <a href="/A023052/a023052.txt">Extended table of values for A023052 and A046074</a>

%H J. Randle, <a href="http://www.jstor.org/stable/3611861">Powerful numbers</a>, Note 3208, Math. Gaz. 52 (1968), 383.

%H J. Randle, <a href="/A000255/a000255.pdf">Powerful numbers</a>, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NarcissisticNumber.html">Narcissistic Number</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>

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

%p P:=proc(q) local a,b,c,k,n,ok; for n from 1 to q do a:=[]; b:=n; ok:=0;

%p for k from 1 to ilog10(n)+1 do if (b mod 10)>1 then ok:=1; fi;

%p a:=[(b mod 10),op(a)]; b:=trunc(b/10); od; b:=0; c:=0;

%p if ok=1 then while c<n do b:=b+1; c:=add(a[k]^b,k=1..nops(a)); od; if

%p c=n then print(n); fi; fi; od; end: P(10^9); # _Paolo P. Lava_, Feb 08 2016

%t 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 *)

%o (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

%Y Cf. A001694, A007532, A005934, A003321, A014576, A046074, A046761, A053540, A161752.

%Y Similar to but different from A005188.

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

%K nonn,base,nice

%O 1,3

%A _David W. Wilson_

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

%E Computed to 10^74 by Xiaoqing Tang

%E A-number typo corrected by _R. J. Mathar_, Jun 22 2009

%E Computed to 10^105 by _Joseph Myers_

%E Cross-references edited by _Joseph Myers_, Jun 28 2009

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)