A320895 a(n) = Sum_{k=1..n} k^3 * tau(k), where tau is A000005.

1, 17, 71, 263, 513, 1377, 2063, 4111, 6298, 10298, 12960, 23328, 27722, 38698, 52198, 72678, 82504, 117496, 131214, 179214, 216258, 258850, 283184, 393776, 440651, 510955, 589687, 721399, 770177, 986177, 1045759, 1242367, 1386115, 1543331, 1714831, 2134735

Offset: 1

Author

Vaclav Kotesovec, Oct 23 2018

Keywords

• nonn

Comments

In general, for m>=0, Sum_{k=1..n} k^m * tau(k) ~ n^(m+1) * ((log(n) + 2*gamma)/(m+1) - 1/(m+1)^2), where gamma is the Euler-Mascheroni constant A001620.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Crossrefs

Cf. A000005, A006218, A143127, A319085.

Programs

Mathematica

Accumulate[Table[k^3*DivisorSigma[0, k], {k, 1, 50}]]

PARI

a(n) = sum(k=1, n, k^3*numdiv(k)); \\ Michel Marcus, Oct 23 2018

Formula

a(n) ~ n^4 * (log(n) + 2*gamma - 1/4)/4, where gamma is the Euler-Mascheroni constant A001620.

a(n) = Sum_{k=1..n} (k^3 / 4) * floor(n/k)^2 * floor(1 + n/k)^2. - Daniel Suteu, Nov 07 2018