A365649 Dirichlet convolution of sigma with Dedekind psi function.

1, 6, 8, 22, 12, 48, 16, 66, 41, 72, 24, 176, 28, 96, 96, 178, 36, 246, 40, 264, 128, 144, 48, 528, 97, 168, 176, 352, 60, 576, 64, 450, 192, 216, 192, 902, 76, 240, 224, 792, 84, 768, 88, 528, 492, 288, 96, 1424, 177, 582, 288, 616, 108, 1056, 288, 1056, 320

Offset

1,2

Links

Table of n, a(n) for n=1..57.

Formula

a(n) = Sum{d|n} A000203(d) * A001615(n/d).

a(p) = A365647(p) + 2 = A365648(p) + 2 where p is a term of A000040.

Multiplicative with a(p^e) = (2 + ((e+1)*p^2 - 2*p - e - 1)*p^e)/(p-1)^2. - Amiram Eldar, Sep 15 2023

From Vaclav Kotesovec, Sep 15 2023: (Start)

Dirichlet g.f.: zeta(s)^2 * zeta(s-1)^2 / zeta(2*s).

Sum_{k=1..n} a(k) ~ 5*n^2 * (log(n)/4 + gamma/2 - 1/8 + 3*zeta'(2)/Pi^2 - 45*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. (End)

Mathematica

f[p_, e_] := (2 + ((e + 1)*p^2 - 2*p - e - 1)*p^e)/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

Prog

(PARI) for(n=1, 100, print1(direuler(p=2, n, (1+X) / ((1-X) * (1 - p*X)^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 15 2023
(Python)
from sympy import divisors, primefactors, prod, reduced_totient, divisor_sigma
def psi(n):
    return n*prod(p+1 for p in primefactors(n))//prod(primefactors(n))
def a(n): return sum(divisor_sigma(d, 1) * psi(n//d) for d in divisors(n))

Crossrefs

Cf. A000040, A000203, A001615, A365647, A365648.

Sequence in context: A242504 A267023 A084962 * A296354 A343759 A024306

Adjacent sequences: A365646 A365647 A365648 * A365650 A365651 A365652

Keyword

nonn,easy,mult

Author

Torlach Rush, Sep 14 2023

Status

approved