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A007433 Inverse Moebius transform applied twice to squares.
(Formerly M4089)
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

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Last modified April 25 16:08 EDT 2019. Contains 322461 sequences. (Running on oeis4.)