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%I M4089 #40 Mar 24 2022 04:11:45
%S 1,6,11,27,27,66,51,112,102,162,123,297,171,306,297,453,291,612,363,
%T 729,561,738,531,1232,678,1026,922,1377,843,1782,963,1818,1353,1746,
%U 1377,2754,1371,2178,1881,3024,1683
%N Inverse Moebius transform applied twice to squares.
%C Dirichlet convolution of A001157 and A000012. Dirichlet convolution of A000005 and A000290 (Jovovic formula). - _R. J. Mathar_, Feb 03 2011
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A007433/b007433.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n) = Sum_{d|n} d^2*tau(n/d). - _Vladeta Jovovic_, Jul 31 2002
%F Equals A134577 * [1, 2, 3, ...]. - _Gary W. Adamson_, Nov 02 2007
%F 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
%F Dirichlet g.f.: zeta(s-2)*zeta(s)^2. - _Benedict W. J. Irwin_, Jul 14 2018
%F 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
%F Sum_{k=1..n} a(k) ~ Zeta(3)^2 * n^3 / 3. - _Vaclav Kotesovec_, Nov 04 2018
%e 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
%t a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* _Robert G. Wilson v_, May 05 2010 *)
%t 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 *)
%o (PARI) /* Dirichlet convolution of A001157, A000012 (Mathar): */
%o a(n)=sumdiv(n, d, sigma(d,2))
%o (PARI) /* Dirichlet convolution of A000005, A000290 (Mathar): */
%o a(n)=sumdiv(n, d, d^2*sigma(n/d,0))
%Y Cf. A134577.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_
%E a(38) corrected by _Ilya Gutkovskiy_, Jan 16 2016
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