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

1, -4, 6, -15, 11, -39, 11, -74, 17, -113, 9, -201, -31, -281, -21, -362, -72, -527, -165, -711, -211, -821, -291, -1141, -490, -1340, -520, -1570, -728, -2028, -1066, -2431, -1211, -2661, -1361, -3272, -1902, -3712, -2012, -4222, -2540, -5040, -3190, -5752, -3386

Offset: 1

Author

Amiram Eldar, Jan 06 2025

Keywords

• sign
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

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)).