A007433 Inverse Moebius transform applied twice to squares.
1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363, 729, 561, 738, 531, 1232, 678, 1026, 922, 1377, 843, 1782, 963, 1818, 1353, 1746, 1377, 2754, 1371, 2178, 1881, 3024, 1683
Offset: 1
Examples
G.f. = x + 6*x^2 + 11*x^3 + 27*x^4 + 27*x^5 + 66*x^6 + 51*x^7 + 112*x^8 + 102*x^9 + ... - _Michael Somos_, Jul 15 2018
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
- N. J. A. Sloane, Transforms
Crossrefs
Cf. A134577.
Programs
-
Mathematica
a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* Robert G. Wilson v, May 05 2010 *) a[ n_] := If[ n < 1, 0, Times @@ (If[ # == 1, 1, (#^(2 #2 + 4) - (#2 + 2) #^2 + #2 + 1) / (#^2 - 1)^2] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 15 2018 *)
-
PARI
/* Dirichlet convolution of A001157, A000012 (Mathar): */ a(n)=sumdiv(n, d, sigma(d,2))
-
PARI
/* Dirichlet convolution of A000005, A000290 (Mathar): */ a(n)=sumdiv(n, d, d^2*sigma(n/d,0))
Formula
a(n) = Sum_{d|n} d^2*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Equals A134577 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} sigma_2(k)*x^k/(1 - x^k), where sigma_2(k) is the sum of squares of divisors of k (A001157). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s-2)*zeta(s)^2. - Benedict W. J. Irwin, Jul 14 2018
a(n) is multiplicative with a(p^e) = (p^(2*e + 4) - (e+2) * p^2 + e+1) / (p^2 - 1)^2. - Michael Somos, Jul 15 2018
Sum_{k=1..n} a(k) ~ Zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Nov 04 2018
Extensions
a(38) corrected by Ilya Gutkovskiy, Jan 16 2016
Comments