A114136 Numbers k such that k divides the sum of digits of all numbers from 1 to k.

1, 3, 5, 7, 9, 18, 21, 24, 38, 58, 78, 98, 298, 498, 501, 537, 698, 702, 707, 711, 716, 898, 1141, 1197, 1501, 1557, 1998, 2703, 2706, 2712, 2715, 3998, 4502, 4509, 4529, 4536, 5998, 7998, 8001, 8097, 9998, 29998, 36001, 36297, 49998, 54504, 54507, 54510

Offset: 1

Author

David W. Wilson, Feb 02 2006

Keywords

• base
• nonn

Comments

a(n) | A037123(a(n)). Sequence is infinite, including 10^(2m)-2 for all m >= 1. For each m >= 1, there is at least one n with A037123(a(n))/a(n) = m.

Number of terms < 10^m for m >= 1: 5, 12, 22, 41, 58, 79, 105, 128, 176, 202, ..., . - Robert G. Wilson v, Jul 06 2018

Certain number forms appear repeatedly: 1(9e)8, 3(9e)8, 5(9e)8, 7(9e)8, 9(9e)8, 2(9o)8, 4(9o)8, 6(9o)8, 8(9o)8; where e represents an even number of times and o an odd number of times. - Robert G. Wilson v, Jul 06 2018

Links

• Giovanni Resta, Table of n, a(n) for n = 1..320 (terms < 5*10^12, first 203 terms from Robert G. Wilson v)

Crossrefs

Cf. A037123.

Programs

Mathematica

lst = {}; k = 1; s = 0; While[k < 10001, If[ Mod[s += Plus @@ IntegerDigits@k, k++] == 0, AppendTo[lst, k -1]]]; lst (* Robert G. Wilson v, Jul 06 2018 *)
nxt[{t_,n_}]:={t+Total[IntegerDigits[n+1]],n+1}; Select[NestList[nxt,{1,1},55000],Divisible[#[[1]],#[[2]]]&][[All,2]] (* Harvey P. Dale, Jan 18 2019 *)

PARI

isok(n) = (sum(k=1, n, sumdigits(k)) % n) == 0; \\ Michel Marcus, Jul 16 2018