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A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

%I A320896 #14 Jan 19 2025 00:35:12
%S A320896 1,9,21,57,77,173,201,329,410,570,614,1046,1098,1322,1562,1962,2030,
%T A320896 2678,2754,3474,3810,4162,4254,5790,6015,6431,6863,7871,7987,9907,
%U A320896 10031,11183,11711,12255,12815,15731,15879,16487,17111,19671,19835,22523,22695,24279
%N A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.
%H A320896 Amiram Eldar, <a href="/A320896/b320896.txt">Table of n, a(n) for n = 1..10000</a>
%H A320896 Ramanujan's Papers, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulas in the analytic theory of numbers</a>, Messenger of Mathematics, XLV, 1916, 81-84.
%F A320896 a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*Pi^8), where g is the Euler-Mascheroni constant A001620, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994, z3 = Zeta'''(2) = A201995 and s1, s2 are the Stieltjes constants, see A082633 and A086279.
%t A320896 Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]
%o A320896 (PARI) a(n) = sum(k=1, n, k*numdiv(k)^2); \\ _Michel Marcus_, Oct 23 2018
%Y A320896 Cf. A061502, A318755, A320897.
%Y A320896 Cf. A000005, A006218, A143127, A319085, A320895.
%K A320896 nonn
%O A320896 1,2
%A A320896 _Vaclav Kotesovec_, Oct 23 2018