A379921 - OEIS

%I #10 Jan 06 2025 10:55:04

%S 1,-4,6,-15,11,-39,11,-74,17,-113,9,-201,-31,-281,-21,-362,-72,-527,

%T -165,-711,-211,-821,-291,-1141,-490,-1340,-520,-1570,-728,-2028,

%U -1066,-2431,-1211,-2661,-1361,-3272,-1902,-3712,-2012,-4222,-2540,-5040,-3190,-5752,-3386

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

%H Amiram Eldar, <a href="/A379921/b379921.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ -zeta(3) * n^3 / 24.

%F 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 Accumulate[Table[(-1)^(k+1) * DivisorSigma[2, k], {k, 1, 100}]]

%o (PARI) list(lim) = {my(s = 0); for(k = 1, lim, s += (-1)^(k+1) * sigma(k, 2); print1(s, ", "));}

%Y Cf. A001157 (sigma_2), A002117, A064602, A068762, A307704.

%K sign,easy

%O 1,2

%A _Amiram Eldar_, Jan 06 2025