A007433 Inverse Moebius transform applied twice to squares. (Formerly M4089)

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

1,2

Comments

Dirichlet convolution of A001157 and A000012. Dirichlet convolution of A000005 and A000290 (Jovovic formula). - R. J. Mathar, Feb 03 2011

Sum of the squares of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

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

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

Example

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

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 *)

Prog

(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))

Crossrefs

Cf. A134577.

Sequence in context: A263419 A140359 A136979 * A061725 A105708 A253908

Adjacent sequences: A007430 A007431 A007432 * A007434 A007435 A007436

Keyword

nonn,mult

Author

N. J. A. Sloane

Extensions

a(38) corrected by Ilya Gutkovskiy, Jan 16 2016

Status

approved