A046913 Sum of divisors of n not congruent to 0 mod 3.

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

Offset

1,2

Links

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

Hershel M. Farkas, On an arithmetical function, Ramanujan J., Vol. 8, No. 3 (2004), pp. 309-315.

Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, The Ramanujan Journal (2020), preprint, arXiv:1905.06506 [math.NT], 2019.

Formula

Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) for p<>3. - Vladeta Jovovic, Sep 11 2002

G.f.: Sum_{k>0} x^k*(1+2*x^k+2*x^(3*k)+x^(4*k))/(1-x^(3*k))^2. - Vladeta Jovovic, Dec 18 2002

a(n) = A000203(3n)-3*A000203(n). - Labos Elemer, Aug 14 2003

Inverse Mobius transform of A091684. - Gary W. Adamson, Jul 03 2008

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^(s-1)). - R. J. Mathar, Feb 10 2011

G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= u^2 + 9 * v^2 + 16 * w^2 - 6 * u*v + 4 * u*w - 24 * v*w - v + w. - Michael Somos, Jul 19 2004

L.g.f.: log(Product_{k>=1} (1 - x^(3*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018

a(n) = A002324(n) + 3*Sum_{j=1, n-1} A002324(j)*A002324(n-j). See Farkas and Guerzhoy links. - Michel Marcus, Jun 01 2019

a(3*n) = a(n). - David A. Corneth, Jun 01 2019

Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 18. - Vaclav Kotesovec, Sep 17 2020

Example

Divisors of 12 are 1 2 3 4 6 12 and discarding 3 6 and 12 we get a(12) = 1 + 2 + 4 = 7.
x + 3*x^2 + x^3 + 7*x^4 + 6*x^5 + 3*x^6 + 8*x^7 + 15*x^8 + x^9 + 18*x^10 + ...

Mathematica

Table[DivisorSigma[1, 3*w]-3*DivisorSigma[1, w], {w, 1, 256}]
DivisorSum[#1, # &, Mod[#, 3] != 0 &] & /@ Range[68] (* Jayanta Basu, Jun 30 2013 *)
f[p_, e_] := If[p == 3, 1, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, sigma(3*n) - 3 * sigma(n))} /* Michael Somos, Jul 19 2004 */
(PARI) a(n) = sigma(n \ 3^valuation(n, 3)) \\ David A. Corneth, Jun 01 2019
(Magma) [SumOfDivisors(3*k)-3*SumOfDivisors(k):k in [1..70]]; // Marius A. Burtea, Jun 01 2019

Crossrefs

Cf. A035191.

Cf. A000726, A051731, A091684, A000203, A002324.

Sequence in context: A323663 A355928 A205298 * A118228 A359079 A245684

Adjacent sequences: A046910 A046911 A046912 * A046914 A046915 A046916

Keyword

nonn,mult

Author

N. J. A. Sloane

Status

approved