A306775 Partial sums of A060648: sum of the inverse Moebius transform of the Dedekind psi function from 1 to n.

1, 5, 10, 20, 27, 47, 56, 78, 95, 123, 136, 186, 201, 237, 272, 318, 337, 405, 426, 496, 541, 593, 618, 728, 765, 825, 878, 968, 999, 1139, 1172, 1266, 1331, 1407, 1470, 1640, 1679, 1763, 1838, 1992, 2035, 2215, 2260, 2390, 2509, 2609, 2658, 2888, 2953, 3101

Offset: 1

Author

Daniel Suteu, Mar 09 2019

Keywords

• nonn

Comments

In general, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) = Sum_{k=1..n} k^m * A064608(floor(n/k)), where psi_m(d) is the generalized Dedekind psi function.

Additionally, for m >= 1, Sum_{k=1..n} Sum_{d|k} psi_m(d) ~ (n^(m+1) * zeta(m+1)^2) / ((m+1) * zeta(2*(m+1))).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Wikipedia, Dedekind psi function

Crossrefs

Cf. A001221, A001615, A034444, A060648, A061503, A064608.

Programs

Maple

with(numtheory): psi := n -> n*mul(1+1/p, p in factorset(n)):
seq(add(psi(i)*floor(n/i), i=1..n), n=1..80); # Ridouane Oudra, Aug 27 2019

Mathematica

Accumulate[Table[Sum[EulerPhi[n/d] * DivisorSigma[0, d^2], {d, Divisors[n]}], {n, 1, 100}]] (* Vaclav Kotesovec, Oct 09 2019 *)

PARI

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

Formula

a(n) ~ (5/4) * n^2.

a(n) = Sum_{k=1..n} A060648(k).

a(n) = Sum_{k=1..n} Sum_{d|k} A001615(d).

a(n) = Sum_{k=1..n} k * A064608(floor(n/k)).

a(n) = (1/2)*Sum_{k=1..n} 2^omega(k) * floor(n/k) * floor(1+n/k).

a(n) = Sum_{k=1..n} A001615(k)*floor(n/k). - Ridouane Oudra, Aug 27 2019