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
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
KEYWORD
nonn,mult
AUTHOR
STATUS
approved