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A320895 a(n) = Sum_{k=1..n} k^3 * tau(k), where tau is A000005.

%I A320895 #15 Nov 08 2018 03:23:19
%S A320895 1,17,71,263,513,1377,2063,4111,6298,10298,12960,23328,27722,38698,
%T A320895 52198,72678,82504,117496,131214,179214,216258,258850,283184,393776,
%U A320895 440651,510955,589687,721399,770177,986177,1045759,1242367,1386115,1543331,1714831,2134735
%N A320895 a(n) = Sum_{k=1..n} k^3 * tau(k), where tau is A000005.
%C A320895 In general, for m>=0, Sum_{k=1..n} k^m * tau(k) ~ n^(m+1) * ((log(n) + 2*gamma)/(m+1) - 1/(m+1)^2), where gamma is the Euler-Mascheroni constant A001620.
%H A320895 Vaclav Kotesovec, <a href="/A320895/b320895.txt">Table of n, a(n) for n = 1..10000</a>
%H A320895 Vaclav Kotesovec, <a href="/A320895/a320895.jpg">Graph - The asymptotic ratio (1000000 terms)</a>
%F A320895 a(n) ~ n^4 * (log(n) + 2*gamma - 1/4)/4, where gamma is the Euler-Mascheroni constant A001620.
%F A320895 a(n) = Sum_{k=1..n} (k^3 / 4) * floor(n/k)^2 * floor(1 + n/k)^2. - _Daniel Suteu_, Nov 07 2018
%t A320895 Accumulate[Table[k^3*DivisorSigma[0, k], {k, 1, 50}]]
%o A320895 (PARI) a(n) = sum(k=1, n, k^3*numdiv(k)); \\ _Michel Marcus_, Oct 23 2018
%Y A320895 Cf. A000005, A006218, A143127, A319085.
%K A320895 nonn
%O A320895 1,2
%A A320895 _Vaclav Kotesovec_, Oct 23 2018