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

1,2

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.

Links

Table of n, a(n) for n=1..34.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A061209, A115518, A257784, A258722.

Sequence in context: A233453 A358442 A136877 * A125904 A272502 A355894

Adjacent sequences: A257966 A257967 A257968 * A257970 A257971 A257972

Keyword

nonn,base

Author

Pieter Post, May 15 2015

Extensions

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

Edited by Jon E. Schoenfield, Mar 01 2022

Status

approved