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A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).

%I A379921 #10 Jan 06 2025 10:55:04
%S A379921 1,-4,6,-15,11,-39,11,-74,17,-113,9,-201,-31,-281,-21,-362,-72,-527,
%T A379921 -165,-711,-211,-821,-291,-1141,-490,-1340,-520,-1570,-728,-2028,
%U A379921 -1066,-2431,-1211,-2661,-1361,-3272,-1902,-3712,-2012,-4222,-2540,-5040,-3190,-5752,-3386
%N A379921 Partial alternating sums of the sigma_2 function: a(n) = Sum_{k=1..n} (-1)^(k+1) * sigma_2(k).
%H A379921 Amiram Eldar, <a href="/A379921/b379921.txt">Table of n, a(n) for n = 1..10000</a>
%F A379921 a(n) ~ -zeta(3) * n^3 / 24.
%F A379921 In general, for m >= 2, Sum_{k=1..n} (-1)^(k+1) * sigma_m(k) ~ -zeta(m+1) * n^(m+1) / ((m+1)*2^(m+1)).
%t A379921 Accumulate[Table[(-1)^(k+1) * DivisorSigma[2, k], {k, 1, 100}]]
%o A379921 (PARI) list(lim) = {my(s = 0); for(k = 1, lim, s += (-1)^(k+1) * sigma(k, 2); print1(s, ", "));}
%Y A379921 Cf. A001157 (sigma_2), A002117, A064602, A068762, A307704.
%K A379921 sign,easy
%O A379921 1,2
%A A379921 _Amiram Eldar_, Jan 06 2025