A365647 Dirichlet convolution of Dedekind psi function with reduced totient function.

1, 4, 6, 11, 10, 24, 14, 26, 26, 40, 22, 64, 26, 56, 56, 58, 34, 104, 38, 106, 78, 88, 46, 148, 74, 104, 102, 148, 58, 224, 62, 128, 122, 136, 128, 272, 74, 152, 144, 244, 82, 312, 86, 232, 232, 184, 94, 326, 146, 296, 188, 274, 106, 408, 200, 340, 210, 232, 118

Offset: 1

Author

Torlach Rush, Sep 14 2023

Keywords

• nonn

Crossrefs

Cf. A000040, A001615, A002322, A365648.

Programs

Mathematica

psi[n_Integer] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Table[DirichletConvolve[psi[k], CarmichaelLambda[k], k, n], {n, 1, 100}] (* Amiram Eldar, Sep 15 2023 *)

Python

from sympy import divisors, primefactors, prod, reduced_totient
def psi(n):
    return n*prod(p+1 for p in primefactors(n))//prod(primefactors(n))
def a(n): return sum(psi(d) * reduced_totient(n//d) for d in divisors(n))

Formula

a(n) = Sum{d|n} A001615(d) * A002322(n/d).

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