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%I #29 Nov 14 2022 01:36:11
%S 1,5,13,21,31,65,57,85,130,155,133,273,183,285,403,341,307,650,381,
%T 651,741,665,553,1105,806,915,1210,1197,871,2015,993,1365,1729,1535,
%U 1767,2730,1407,1905,2379,2635,1723,3705,1893,2793,4030,2765,2257,4433,2850,4030
%N a(n) = Sum_{d|n} d^2*A000593(n/d).
%C Multiplicative because this sequence is the Dirichlet convolution of A000290 and A000593 which are both multiplicative. - _Andrew Howroyd_, Jul 27 2018
%H Seiichi Manyama, <a href="/A288418/b288418.txt">Table of n, a(n) for n = 1..10000</a>
%F L.g.f.: log(Product_{k>=1} (1 + x^k)^sigma(k)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, Jun 19 2018
%F From _Amiram Eldar_, Nov 13 2022: (Start)
%F a(n) = A001001(n) for odd n.
%F Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(e+1)-1)*(p^(e+2)-1)/((p-1)*(p^2-1)) for p > 2.
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(2)*zeta(3)/4 = A183699 / 4 = 0.494326... . (End)
%t a[n_] := DivisorSum[n, Function[d, d^2*DivisorSum[n/d, If[OddQ[#], #, 0]&]] ];
%t Array[a, 50] (* _Jean-François Alcover_, Jul 03 2017 *)
%t f[p_, e_] := (p^(e + 1) - 1)*(p^(e + 2) - 1)/((p - 1)*(p^2 - 1)); f[2, e_] := (4^(e + 1) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 13 2022 *)
%o (PARI) a(n) = sumdiv(n, d, d^2*sigma((n/d)>>valuation(n/d, 2))); \\ _Michel Marcus_, Jul 03 2017; corrected Jun 12 2022
%Y Cf. A000290, A001001, A192065, A183699.
%Y Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), A109386 (k=1), this sequence (k=2), A288419 (k=3), A288420 (k=4).
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, Jun 09 2017
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