A284099 a(n) = Sum_{d|n, d == 1 (mod 7)} d.

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 16, 9, 1, 1, 1, 1, 1, 23, 1, 9, 1, 1, 1, 1, 30, 16, 1, 9, 1, 1, 1, 37, 1, 1, 1, 9, 1, 1, 44, 23, 16, 1, 1, 9, 1, 51, 1, 1, 1, 1, 1, 9, 58, 30, 1, 16, 1, 1, 1, 73, 1, 23, 1, 1, 1, 1, 72, 45, 1, 1, 16, 1, 1, 79, 1, 9, 1, 1

Offset: 1

Author

Seiichi Manyama, Mar 20 2017

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A109703.

Cf. Sum_{d|n, d == 1 (mod k)} d: A000593 (k=2), A078181 (k=3), A050449 (k=4), A284097 (k=5), A284098 (k=6), this sequence (k=7), A284100 (k=8).

Cf. Sum_{d|n, d == k (mod 7)} d: this sequence (k=1), A284443 (k=2), A284444 (k=3), A284445 (k=4), A284446 (k=5), A284105 (k=6).

Programs

Mathematica

Table[Sum[If[Mod[d, 7] == 1, d, 0], {d, Divisors[n]}], {n, 82}] (* Indranil Ghosh, Mar 21 2017 *)
Table[DivisorSum[n,#&,Mod[#,7]==1&],{n,90}] (* Harvey P. Dale, Aug 08 2021 *)

PARI

for(n=1, 82, print1(sumdiv(n, d, if(Mod(d, 7)==1, d, 0)), ", ")) \\ Indranil Ghosh, Mar 21 2017

Python

from sympy import divisors
def a(n): return sum([d for d in divisors(n) if d%7==1]) # Indranil Ghosh, Mar 21 2017

Formula

G.f.: Sum_{k>=0} (7*k + 1)*x^(7*k+1)/(1 - x^(7*k+1)). - Ilya Gutkovskiy, Mar 21 2017

G.f.: Sum_{n >= 1} x^n*(1 + 6*x^(7*n))/(1 - x^(7*n))^2. - Peter Bala, Dec 19 2021

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/84 = 0.117495... . - Amiram Eldar, Nov 26 2023