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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.

%I A257969 #32 Mar 03 2022 05:14:50
%S A257969 1,10,100,1000,7972,10000,53941,79720,100000,134242,539410,698614,
%T A257969 797200,1000000,1342420,5394100,6986140,7525615,7972000,9000864,
%U A257969 10000000,10057054,13424200,15366307,17513566,20602674,23280211,24716905,25274655,25665559,32083981,34326702,34446204,34534816
%N 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.
%C A257969 All powers of 10 are terms of this sequence.
%C A257969 If m is a term, then so is 10*m.
%C A257969 Number of terms < 10^k for k >= 1: 1, 2, 3, 5, 8, 13, 20, 62.
%F A257969 {m : sod(m^(k+1)) - sod(m^k) = f for k=1..8}.
%e A257969 7972 is in the sequence, because the difference between the successive sum-of-digit values is 15:
%e A257969   sod(7972) = 25;
%e A257969   sod(7972^2) = 40;
%e A257969   sod(7972^3) = 55;
%e A257969   sod(7972^4) = 70;
%e A257969   sod(7972^5) = 85;
%e A257969   sod(7972^6) = 100;
%e A257969   sod(7972^7) = 115;
%e A257969   sod(7972^8) = 130;
%e A257969   sod(7972^9) = 145;
%e A257969   sod(7972^10) = 178, where the increment is no longer 15.
%e A257969 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.
%e A257969   sod(134242) = 16;
%e A257969   sod(134242^2) = 40;
%e A257969   sod(134242^3) = 64;
%e A257969   sod(134242^4) = 88;
%e A257969   sod(134242^5) = 112;
%e A257969   sod(134242^6) = 136;
%e A257969   sod(134242^7) = 160;
%e A257969   sod(134242^8) = 184;
%e A257969   sod(134242^9) = 208;
%e A257969   sod(134242^10) = 232;
%e A257969   sod(134242^11) = 283, where the increment is no longer 24.
%t A257969 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 *)
%t A257969 Select[Range[35*10^6],Length[Union[Differences[Total/@IntegerDigits[ #^Range[9]]]]] ==1&] (* _Harvey P. Dale_, Aug 23 2017 *)
%o A257969 (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
%Y A257969 Cf. A061209, A115518, A257784, A258722.
%K A257969 nonn,base
%O A257969 1,2
%A A257969 _Pieter Post_, May 15 2015
%E A257969 Corrected and extended by _Harvey P. Dale_, Aug 23 2017
%E A257969 Edited by _Jon E. Schoenfield_, Mar 01 2022