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

Author

Torlach Rush, Sep 14 2023

Keywords

• nonn
• easy
• mult

Crossrefs

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

Programs

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 *)

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))

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)