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

1,2

Links

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

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

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 *)

Prog

(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)

Crossrefs

Cf. A000203, A035191, A046913, A346933.

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

Cf. A326399, A326400.

Sequence in context: A175039 A222405 A146970 * A096273 A069981 A000199

Adjacent sequences: A078705 A078706 A078707 * A078709 A078710 A078711

Keyword

mult,easy,nonn

Author

Vladeta Jovovic, Dec 18 2002

Extensions

Extended by Klaus Brockhaus and Benoit Cloitre, Dec 20 2002

Status

approved