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

%I A316312 #42 May 22 2021 04:26:34
%S A316312 1,3,5,7,9,12,15,20,27,40,45,60,63,80,81,100,180,181,300,360,363,500,
%T A316312 540,545,700,720,727,900,909,912,915,1137,1140,1200,1500,1560,1563,
%U A316312 2000,2700,2720,2727,4000,4500,4540,4545,6000,6300,6360,6363,8000,8100,8180
%N A316312 Numbers k such that the sum of the digits of the numbers 1, 2, 3, ... up to (k - 1) is divisible by k.
%C A316312 Numbers k such that A007953(A007908(k - 1)) is divisible by k. - _Felix Fröhlich_, Jun 29 2018
%C A316312 From _Robert Israel_, Jun 29 2018: (Start)
%C A316312 Numbers k such that A037123(k - 1) is divisible by k.
%C A316312 If m is even, then 10^m, 3 * 10^m, 5 * 10^m, 7 * 10^m and 9 * 10^m are included.
%C A316312 If m is odd, then 2 * 10^m, 4 * 10^m, 6 * 10^m, and 8 * 10^m are included. (End)
%C A316312 Is it true that if k is a term then 100 * k is a term?
%H A316312 Henry Bottomley, <a href="/A316312/b316312.txt">Table of n, a(n) for n = 1..118</a>
%e A316312 For n = 7, sum of the digits of the numbers 1 to 6 is 21, which is divisible by 7.
%e A316312 For n = 12, sum of the digits of the numbers 1 to 11 is 48, which is divisible by 12.
%e A316312 For n = 15, sum of the digits of the numbers 1 to 14 is 60, which is divisible by 15.
%e A316312 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.
%p A316312 t:= 0: Res:= NULL:
%p A316312 for n from 1 to 10000 do
%p A316312   t:= t + convert(convert(n-1,base,10),`+`);
%p A316312   if (t/n)::integer then Res:= Res, n fi
%p A316312 od:
%p A316312 Res; # _Robert Israel_, Jun 29 2018
%t A316312 s = 0; Reap[Do[If[Mod[s, n] == 0, Sow[n]]; s += Plus @@ IntegerDigits@n, {n, 10000}]][[2, 1]] (* _Giovanni Resta_, Jun 29 2018 *)
%o A316312 (PARI) sumsod(n) = sum(i=1, n, sumdigits(i))
%o A316312 is(n) = sumsod(n-1)%n==0 \\ _Felix Fröhlich_, Jun 29 2018
%o A316312 (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
%Y A316312 Cf. A007953, A007908, A037123, A110740, A114136.
%K A316312 nonn,base
%O A316312 1,2
%A A316312 _Debapriyay Mukhopadhyay_, Jun 29 2018
%E A316312 More terms from _Felix Fröhlich_, Jun 29 2018