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
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.
Programs
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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
Comments