A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.

1, 3, 5, 7, 9, 12, 15, 20, 27, 40, 45, 60, 63, 80, 81, 100, 180, 181, 300, 360, 363, 500, 540, 545, 700, 720, 727, 900, 909, 912, 915, 1137, 1140, 1200, 1500, 1560, 1563, 2000, 2700, 2720, 2727, 4000, 4500, 4540, 4545, 6000, 6300, 6360, 6363, 8000, 8100, 8180

Offset: 1

Author

Debapriyay Mukhopadhyay, Jun 29 2018

Keywords

• nonn
• base

Comments

Numbers k such that A007953(A007908(k - 1)) is divisible by k. - Felix Fröhlich, Jun 29 2018

From Robert Israel, Jun 29 2018: (Start)

Numbers k such that A037123(k - 1) is divisible by k.

If m is even, then 10^m, 3 * 10^m, 5 * 10^m, 7 * 10^m and 9 * 10^m are included.

If m is odd, then 2 * 10^m, 4 * 10^m, 6 * 10^m, and 8 * 10^m are included. (End)

Is it true that if k is a term then 100 * k is a term?

Examples

			For n = 7, sum of the digits of the numbers 1 to 6 is 21, which is divisible by 7.
For n = 12, sum of the digits of the numbers 1 to 11 is 48, which is divisible by 12.
For n = 15, sum of the digits of the numbers 1 to 14 is 60, which is divisible by 15.
16 is not in the sequence because the sum of the digits of the numbers 1 to 15 is 66, which is not divisible by 16.
		

Links

• Henry Bottomley, Table of n, a(n) for n = 1..118

Crossrefs

Cf. A007953, A007908, A037123, A110740, A114136.

Programs

Maple

t:= 0: Res:= NULL:
for n from 1 to 10000 do
  t:= t + convert(convert(n-1,base,10),`+`);
  if (t/n)::integer then Res:= Res, n fi
od:
Res; # Robert Israel, Jun 29 2018

Mathematica

s = 0; Reap[Do[If[Mod[s, n] == 0, Sow[n]]; s += Plus @@ IntegerDigits@n, {n, 10000}]][[2, 1]] (* Giovanni Resta, Jun 29 2018 *)

PARI

sumsod(n) = sum(i=1, n, sumdigits(i))
is(n) = sumsod(n-1)%n==0 \\ Felix Fröhlich, Jun 29 2018

PARI

upto(n) = my(s=0,res=List()); for(i=0, n, s += vecsum(digits(i)); if(s%(i+1)==0, listput(res, i+1))); res \\ David A. Corneth, Jun 29 2018

Extensions

More terms from Felix Fröhlich, Jun 29 2018