A260768 - OEIS

%I #50 Sep 08 2015 03:51:05

%S 0,1,2,3,4,5,6,7,8,9,10,12,15,18,24,27,30,54,57,66,93,100,107,110,111,

%T 120,125,138,143,159,168,170,179,225,243,261,300,309,338,339,347,354,

%U 381,438,441,501,521,528,534,552,567,573,576,593,645,661,709,724,738,807,849,903,926,927

%N Numbers n such that n equals the sum of digit_sum(n^p) for p = 1 to some k>=1, where digit_sum = A007953.

%C 'digit_sum' is the 'sum of the digits' as defined in A007953.

%C The number of terms < 10^k: 9, 20, 63, 160, 454, 1333, 3704, ..., .

%C So far, 3705 terms, 70.93% are congruent to 0 (mod 3), 8.26% congruent to 1 (mod 3) and 20.81% congruent to 2 (mod 3).

%H Pieter Post and Robert G. Wilson v, <a href="/A260768/b260768.txt">Table of n, a(n) for n = 1..3705</a>

%F All numbers of the form 10^p are members; for n = 1-9, a(n)=n are trivial solutions.

%e 57 is in the sequence because digit_sum(57) + digit_sum(57^2) + digit_sum(57^3) = 12 + 18 + 27 = 57. In this example, k is 3.

%p filter:= proc(n)

%p   local t,p;

%p   t:= 0;

%p   for p from 1 while t < n do

%p     t:= t+ sod(n^p);

%p   od:

%p   evalb(t = n)

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Aug 16 2015

%t fQ[n_] := If[ IntegerQ@ Log10@ n, True, Block[{pwr = 1, s = 0}, While[s = s + Plus @@ IntegerDigits[n^pwr]; s < n, pwr++]; s == n]]; Select[ Range[0, 1000], fQ]

%o (PARI) is(n)=my(s); for(p=1,n,s+=sumdigits(n^p); if(s>=n, return(s==n))) \\ _Charles R Greathouse IV_, Aug 07 2015

%Y Cf. A007953, A259313.

%K nonn,base

%O 1,3

%A _Pieter Post_ and _Robert G. Wilson v_, Jul 30 2015