A260768 Numbers n such that n equals the sum of digit_sum(n^p) for p = 1 to some k>=1, where digit_sum = A007953.

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, 120, 125, 138, 143, 159, 168, 170, 179, 225, 243, 261, 300, 309, 338, 339, 347, 354, 381, 438, 441, 501, 521, 528, 534, 552, 567, 573, 576, 593, 645, 661, 709, 724, 738, 807, 849, 903, 926, 927

Offset: 1

Author

Pieter Post and Robert G. Wilson v, Jul 30 2015

Keywords

• nonn
• base

Comments

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

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

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).

Examples

			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.
		

Links

• Pieter Post and Robert G. Wilson v, Table of n, a(n) for n = 1..3705

Crossrefs

Cf. A007953, A259313.

Programs

Maple

filter:= proc(n)
  local t,p;
  t:= 0;
  for p from 1 while t < n do
    t:= t+ sod(n^p);
  od:
  evalb(t = n)
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 16 2015

Mathematica

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]

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

Formula

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