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

1,2

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)

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A006218, A143127, A319085.

Sequence in context: A141959 A347334 A290243 * A288748 A069496 A047978

Adjacent sequences: A320892 A320893 A320894 * A320896 A320897 A320898

Keyword

nonn

Author

Vaclav Kotesovec, Oct 23 2018

Status

approved