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A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.

%I A320897 #16 Jan 19 2025 00:35:03
%S A320897 1,17,53,197,297,873,1069,2093,2822,4422,4906,10090,10766,13902,17502,
%T A320897 23902,25058,36722,38166,52566,59622,67366,69482,106346,111971,122787,
%U A320897 134451,162675,166039,223639,227483,264347,281771,300267,319867,424843,430319,453423
%N A320897 a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.
%C A320897 In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).
%H A320897 Amiram Eldar, <a href="/A320897/b320897.txt">Table of n, a(n) for n = 1..10000</a>
%H A320897 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 A320897 a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*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 A320897 Accumulate[Table[k^2*DivisorSigma[0, k]^2, {k, 1, 50}]]
%o A320897 (PARI) a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ _Michel Marcus_, Oct 23 2018
%Y A320897 Cf. A061502, A318755, A320896.
%Y A320897 Cf. A000005, A006218, A143127, A319085, A320895.
%K A320897 nonn
%O A320897 1,2
%A A320897 _Vaclav Kotesovec_, Oct 23 2018