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A320897
a(n) = Sum_{k=1..n} k^2 * tau(k)^2, where tau is A000005.
2
1, 17, 53, 197, 297, 873, 1069, 2093, 2822, 4422, 4906, 10090, 10766, 13902, 17502, 23902, 25058, 36722, 38166, 52566, 59622, 67366, 69482, 106346, 111971, 122787, 134451, 162675, 166039, 223639, 227483, 264347, 281771, 300267, 319867, 424843, 430319, 453423
OFFSET
1,2
COMMENTS
In general, for m>=0, Sum_{k=1..n} k^m * tau(k)^2 ~ n^(m+1) * (log(n))^3 / ((m+1) * Pi^2).
LINKS
Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84.
FORMULA
a(n) ~ n^3 * (2*Pi^6*(-1 + 12*g - 54*g^2 + 108*g^3 + 36*s1 - 324*g*s1 + 54*s2) - 93312*z1^3 + 2592*Pi^2*z1*(-z1 + 12*g*z1 + 6*z2) - 72*Pi^4*(z1 - 12*g*z1 + 54*g^2*z1 - 36*s1*z1 - 3*z2 + 36*g*z2 + 6*z3) + 6*(Pi^6*(1 - 12*g + 54*g^2 - 36*s1) + 1296*Pi^2*z1^2 - 36*Pi^4*(-z1 + 12*g*z1 + 3*z2))*log(n) + 9*((-1 + 12*g)*Pi^6 - 36*Pi^4*z1)*log(n)^2 + 9*Pi^6*log(n)^3) / (27*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^2*DivisorSigma[0, k]^2, {k, 1, 50}]]
PROG
(PARI) a(n) = sum(k=1, n, k^2*numdiv(k)^2); \\ Michel Marcus, Oct 23 2018
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 23 2018
STATUS
approved