A320896 a(n) = Sum_{k=1..n} k * tau(k)^2, where tau is A000005.

1, 9, 21, 57, 77, 173, 201, 329, 410, 570, 614, 1046, 1098, 1322, 1562, 1962, 2030, 2678, 2754, 3474, 3810, 4162, 4254, 5790, 6015, 6431, 6863, 7871, 7987, 9907, 10031, 11183, 11711, 12255, 12815, 15731, 15879, 16487, 17111, 19671, 19835, 22523, 22695, 24279

Offset

1,2

Links

Table of n, a(n) for n=1..44.

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.

Formula

a(n) ~ n^2 * (3*(Pi^6*(-1 - 24*g^2 + 32*g^3 + g*(8 - 96*s1) + 16*s1 + 16*s2) - 13824*z1^3 + 576*Pi^2*z1*((-1 + 8*g)*z1 + 4*z2) - 8*Pi^4*(3*(1 - 8*g + 24*g^2 - 16*s1)*z1 - 6*z2 + 48*g*z2 + 8*z3)) + 6*(Pi^6*(1 - 8*g + 24*g^2 - 16*s1) + 576*Pi^2*z1^2 - 24*Pi^4*(-z1 + 8*g*z1 + 2*z2))*log(n) + 6*((-1 + 8*g)*Pi^6 - 24*Pi^4*z1)*log(n)^2 + 4*Pi^6*log(n)^3) / (8*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.

Mathematica

Accumulate[Table[k*DivisorSigma[0, k]^2, {k, 1, 50}]]

Prog

(PARI) a(n) = sum(k=1, n, k*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018

Crossrefs

Cf. A061502, A318755, A320897.

Cf. A000005, A006218, A143127, A319085, A320895.

Sequence in context: A147169 A216167 A246327 * A127989 A147337 A020290

Adjacent sequences: A320893 A320894 A320895 * A320897 A320898 A320899

Keyword

nonn

Author

Vaclav Kotesovec, Oct 23 2018

Status

approved