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A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

%I A282548 #31 Nov 21 2024 05:30:44
%S A282548 0,1,4098,531444,16785412,244140630,2177857512,13841287208,
%T A282548 68753047560,282431130813,1000488301740,3138428376732,8920506494928,
%U A282548 23298085122494,56721594978384,129747072969720,281612482805776,582622237229778,1157402774071674
%N A282548 Expansion of phi_{12, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
%C A282548 Multiplicative because A013959 is. - _Andrew Howroyd_, Jul 25 2018
%H A282548 Seiichi Manyama, <a href="/A282548/b282548.txt">Table of n, a(n) for n = 0..10000</a>
%F A282548 a(n) = n*A013959(n) for n > 0.
%F A282548 a(n) = (441*A282549(n) + 250*A282576(n) - 691*A058550(n))/65520.
%F A282548 Sum_{k=1..n} a(k) ~ zeta(12) * n^13 / 13. - _Amiram Eldar_, Sep 06 2023
%F A282548 From _Amiram Eldar_, Oct 30 2023: (Start)
%F A282548 Multiplicative with a(p^e) = p^e * (p^(11*e+11)-1)/(p^11-1).
%F A282548 Dirichlet g.f.: zeta(s-1)*zeta(s-12). (End)
%t A282548 Table[n * DivisorSigma[11, n], {n, 0, 18}] (* _Amiram Eldar_, Sep 06 2023 *)
%o A282548 (PARI) a(n) = if(n < 1, 0, n*sigma(n, 11)) \\ _Andrew Howroyd_, Jul 25 2018
%Y A282548 Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), this sequence (phi_{12, 1}).
%Y A282548 Cf. A282549 (E_2*E_4^3), A282576 (E_2*E_6^2), A058550 (E_14).
%Y A282548 Cf. A013670.
%K A282548 nonn,easy,mult
%O A282548 0,3
%A A282548 _Seiichi Manyama_, Feb 18 2017