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

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Examples

			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 + ...
		

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.

Crossrefs

Cf. A035191.

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

Programs

Magma

[SumOfDivisors(3*k)-3*SumOfDivisors(k):k in [1..70]]; // Marius A. Burtea, Jun 01 2019

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 *)

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

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