A284098 - OEIS

%I #28 Nov 26 2023 03:11:02

%S 1,1,1,1,1,1,8,1,1,1,1,1,14,8,1,1,1,1,20,1,8,1,1,1,26,14,1,8,1,1,32,1,

%T 1,1,8,1,38,20,14,1,1,8,44,1,1,1,1,1,57,26,1,14,1,1,56,8,20,1,1,1,62,

%U 32,8,1,14,1,68,1,1,8,1,1,74,38,26,20,8,14,80,1,1

%N a(n) = Sum_{d|n, d == 1 (mod 6)} d.

%H Seiichi Manyama, <a href="/A284098/b284098.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=0} (6*k + 1)*x^(6*k+1)/(1 - x^(6*k+1)). - _Ilya Gutkovskiy_, Mar 21 2017

%F G.f.: Sum_{n >= 1} x^n*(1 + 5*x^(6*n))/(1 - x^(6*n))^2. - _Peter Bala_, Dec 19 2021

%F Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/72 = 0.137077... (A086729). - _Amiram Eldar_, Nov 26 2023

%t Table[Sum[If[Mod[d, 6] == 1, d, 0], {d, Divisors[n]}], {n, 80}] (* _Indranil Ghosh_, Mar 21 2017 *)

%o (PARI) for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 6)==1, d, 0)), ", ")) \\ _Indranil Ghosh_, Mar 21 2017

%o (Python)

%o from sympy import divisors

%o def a(n): return sum([d for d in divisors(n) if d%6==1]) # _Indranil Ghosh_, Mar 21 2017

%Y Cf. A086729, A109701.

%Y Cf. Sum_{d|n, d==1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), this sequence (k=6), A284099 (k=7), A284100 (k=8).

%K nonn,easy

%O 1,7

%A _Seiichi Manyama_, Mar 20 2017