A078708 Sum of divisors d of n such that n/d is not congruent to 0 mod 3.

1, 3, 3, 7, 6, 9, 8, 15, 9, 18, 12, 21, 14, 24, 18, 31, 18, 27, 20, 42, 24, 36, 24, 45, 31, 42, 27, 56, 30, 54, 32, 63, 36, 54, 48, 63, 38, 60, 42, 90, 42, 72, 44, 84, 54, 72, 48, 93, 57, 93, 54, 98, 54, 81, 72, 120, 60, 90, 60, 126, 62, 96, 72, 127, 84, 108, 68, 126, 72, 144

Offset: 1

Author

Vladeta Jovovic, Dec 18 2002

Keywords

• mult
• easy
• nonn

Links

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

Crossrefs

Cf. A000203, A035191, A046913, A346933.

Cf. A002131 (k=2), this sequence (k=3), A285895 (k=4), A285896 (k=5).

Cf. A326399, A326400.

Programs

Mathematica

f[p_, e_] := If[p == 3, 3^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

PARI

for(n=1,70,d=divisors(n); s=0; for(j=1,matsize(d)[2],if((n/d[j])%3>0,s=s+d[j])); print1(s,","))

PARI

a(n)=sumdiv(n,d,if((n/d)%3,1,0)*d)

Formula

G.f.: Sum_{k>0} x^k*(1+x^k)^2*(1+x^(2*k))/(1-x^(3*k))^2.

a(n) = (A000203(3*n)-A000203(n))/3. - Vladeta Jovovic, Dec 22 2003

G.f.: Sum_{k>=1} k*x^k*(1 + x^k)/(1 - x^(3*k)). - Ilya Gutkovskiy, Sep 13 2019

From R. J. Mathar, May 25 2020: (Start)

a(n) = A326399(n) + A326400(n).

a(n) = A000203(n) - A000203(n/3), where A000203(.) = 0 for non-integer arguments. (End)

From Amiram Eldar, Oct 30 2022: (Start)

Multiplicative with a(3^e) = 3^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 3.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/27 = 0.731081... (A346933). (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^s). - Amiram Eldar, Dec 30 2022

Extensions

Extended by Klaus Brockhaus and Benoit Cloitre, Dec 20 2002