A257969 Numbers m such that the sum of the digits (sod) of m, m^2, m^3, ..., m^9 are in arithmetic progression: sod(m^(k+1)) - sod(m^k) = f for k=1..8.

1, 10, 100, 1000, 7972, 10000, 53941, 79720, 100000, 134242, 539410, 698614, 797200, 1000000, 1342420, 5394100, 6986140, 7525615, 7972000, 9000864, 10000000, 10057054, 13424200, 15366307, 17513566, 20602674, 23280211, 24716905, 25274655, 25665559, 32083981, 34326702, 34446204, 34534816

Offset: 1

Author

Pieter Post, May 15 2015

Keywords

• nonn
• base

Comments

All powers of 10 are terms of this sequence.

If m is a term, then so is 10*m.

Number of terms < 10^k for k >= 1: 1, 2, 3, 5, 8, 13, 20, 62.

Examples

			7972 is in the sequence, because the difference between the successive sum-of-digit values is 15:
  sod(7972) = 25;
  sod(7972^2) = 40;
  sod(7972^3) = 55;
  sod(7972^4) = 70;
  sod(7972^5) = 85;
  sod(7972^6) = 100;
  sod(7972^7) = 115;
  sod(7972^8) = 130;
  sod(7972^9) = 145;
  sod(7972^10) = 178, where the increment is no longer 15.
But there are seven numbers below 10^9 with a longer sequence (namely, 134242, 23280211, 40809168, 46485637, 59716223, 66413917, and 97134912) where sod(m^(k+1)) - sod(m^k) = f for k=1..9.
  sod(134242) = 16;
  sod(134242^2) = 40;
  sod(134242^3) = 64;
  sod(134242^4) = 88;
  sod(134242^5) = 112;
  sod(134242^6) = 136;
  sod(134242^7) = 160;
  sod(134242^8) = 184;
  sod(134242^9) = 208;
  sod(134242^10) = 232;
  sod(134242^11) = 283, where the increment is no longer 24.
		

Crossrefs

Cf. A061209, A115518, A257784, A258722.

Programs

Mathematica

fQ[n_] := Block[{g}, g[x_] := Power[x, #] & /@ Range@ 9; Length@ DeleteDuplicates@ Differences[Total[IntegerDigits@ #] & /@ g@ n] == 1]; Select[Range@ 1000000, fQ] (* Michael De Vlieger, Jun 12 2015 *)
Select[Range[35*10^6],Length[Union[Differences[Total/@IntegerDigits[ #^Range[9]]]]] ==1&] (* Harvey P. Dale, Aug 23 2017 *)

PARI

isok(n) = {my(osod = sumdigits(n^2)); my(f = osod - sumdigits(n)); for (k=3, 9, my(nsod = sumdigits(n^k)); if (nsod - osod != f, return (0)); osod = nsod;); return (1);} \\ Michel Marcus, May 28 2015

Formula

{m : sod(m^(k+1)) - sod(m^k) = f for k=1..8}.

Extensions

Corrected and extended by Harvey P. Dale, Aug 23 2017

Edited by Jon E. Schoenfield, Mar 01 2022