%I #53 May 17 2023 04:04:39

%S 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,

%T 56,30,18,32,63,12,54,48,7,38,60,14,90,42,24,44,84,6,72,48,31,57,93,

%U 18,98,54,3,72,120,20,90,60,42,62,96,8,127,84,36,68,126

%N Sum of divisors of n not congruent to 0 mod 3.

%H Seiichi Manyama, <a href="/A046913/b046913.txt">Table of n, a(n) for n = 1..10000</a>

%H Hershel M. Farkas, <a href="https://doi.org/10.1007/s11139-004-0141-5">On an arithmetical function</a>, Ramanujan J., Vol. 8, No. 3 (2004), pp. 309-315.

%H Pavel Guerzhoy and Ka Lun Wong, <a href="https://doi.org/10.1007/s11139-020-00296-5">Farkas' identities with quartic characters</a>, The Ramanujan Journal (2020), <a href="https://arxiv.org/abs/1905.06506">preprint</a>, arXiv:1905.06506 [math.NT], 2019.

%F Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) for p<>3. - _Vladeta Jovovic_, Sep 11 2002

%F 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

%F a(n) = A000203(3n)-3*A000203(n). - _Labos Elemer_, Aug 14 2003

%F Inverse Mobius transform of A091684. - _Gary W. Adamson_, Jul 03 2008

%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/3^(s-1)). - _R. J. Mathar_, Feb 10 2011

%F 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

%F 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

%F 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

%F a(3*n) = a(n). - _David A. Corneth_, Jun 01 2019

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 18. - _Vaclav Kotesovec_, Sep 17 2020

%e Divisors of 12 are 1 2 3 4 6 12 and discarding 3 6 and 12 we get a(12) = 1 + 2 + 4 = 7.

%e 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 + ...

%t Table[DivisorSigma[1, 3*w]-3*DivisorSigma[1, w], {w, 1, 256}]

%t DivisorSum[#1, # &, Mod[#, 3] != 0 &] & /@ Range[68] (* _Jayanta Basu_, Jun 30 2013 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, 0, sigma(3*n) - 3 * sigma(n))} /* _Michael Somos_, Jul 19 2004 */

%o (PARI) a(n) = sigma(n \ 3^valuation(n, 3)) \\ _David A. Corneth_, Jun 01 2019

%o (Magma) [SumOfDivisors(3*k)-3*SumOfDivisors(k):k in [1..70]]; // _Marius A. Burtea_, Jun 01 2019

%Y Cf. A035191.

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

%K nonn,mult

%O 1,2

%A _N. J. A. Sloane_