A321879 Partial sums of the Jordan function J_2(k), for 1 <= k <= n.

0, 1, 4, 12, 24, 48, 72, 120, 168, 240, 312, 432, 528, 696, 840, 1032, 1224, 1512, 1728, 2088, 2376, 2760, 3120, 3648, 4032, 4632, 5136, 5784, 6360, 7200, 7776, 8736, 9504, 10464, 11328, 12480, 13344, 14712, 15792, 17136, 18288, 19968, 21120, 22968, 24408

Offset: 0

Author

Daniel Suteu, Nov 20 2018

Keywords

• nonn
• easy

Comments

In general, for m >= 1, Sum_{k=1..n} J_m(k) = Sum_{k=1..n} mu(k) * (Bernoulli(m+1, 1+floor(n/k)) - Bernoulli(m+1, 0)) / (m+1), where mu(k) is the Moebius function and Bernoulli(n,x) are the Bernoulli polynomials.

In general, for m >= 1, Sum_{k=1..n} J_m(k) ~ n^(m+1) / ((m+1) * zeta(m+1)).

In general, for m >= 1, Sum_{k=1..n} J_m(k) = Sum_{k=1..n} k^m*A002321(floor(n/k)). - Ridouane Oudra, Jul 03 2025

Links

• Wikipedia, Jordan's totient function

Crossrefs

Cf. A002088, A007434, A008683, A002321.

Programs

Mathematica

a[n_]:= Sum[MoebiusMu[k]*BernoulliB[3,1+Floor[n/k]]/3, {k,1,n}]; Array[a, 50, 0] (* Stefano Spezia, Nov 21 2018 *)

PARI

a(n) = sum(k=1, n, moebius(k) * ((n\k)^3/3 + (n\k)^2/2 + (n\k)/6));

Formula

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

a(n) = Sum_{k=1..n} A007434(k).

a(n) = Sum_{k=1..n} mu(k) * Bernoulli(3, 1+floor(n/k)) / 3, where mu(k) is the Moebius function and Bernoulli(n,x) are the Bernoulli polynomials.

a(n) = Sum_{k=1..n} k^2*A002321(floor(n/k)). - Ridouane Oudra, Jul 03 2025