A347092 Dirichlet inverse of A322577, which is the convolution of Dedekind psi with Euler phi.

1, -4, -6, 5, -10, 24, -14, -4, 10, 40, -22, -30, -26, 56, 60, 5, -34, -40, -38, -50, 84, 88, -46, 24, 26, 104, -6, -70, -58, -240, -62, -4, 132, 136, 140, 50, -74, 152, 156, 40, -82, -336, -86, -110, -100, 184, -94, -30, 50, -104, 204, -130, -106, 24, 220, 56, 228, 232, -118, 300, -122, 248, -140, 5, 260, -528, -134

Offset: 1

Author

Antti Karttunen, Aug 18 2021

Keywords

• sign
• easy
• mult

Comments

Multiplicative because A322577 is.

Links

• Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001615, A322577, A347093.

Cf. A023900, A323363.

Programs

Haskell

import Math.NumberTheory.Primes
a n = product . map (\(p, e) -> if even e then 1 + unPrime p^2 else -2*unPrime p) . factorise $ n -- Sebastian Karlsson, Oct 29 2021

Mathematica

f[p_, e_] := If[EvenQ[e], p^2 + 1, -2*p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

PARI

up_to = 16384;
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA322577(n)));
A347092(n) = v347092[n];

PARI

A347092(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]%2, -2*f[i, 1], 1+(f[i, 1]^2))); }; \\ (after Sebastian Karlsson's multiplicative formula) - Antti Karttunen, Nov 11 2021

Python

from sympy import factorint, prod
def f(p, e): return 1 + p**2 if e%2 == 0 else -2*p
def a(n):
    factors = factorint(n)
    return prod(f(p, factors[p]) for p in factors) # Sebastian Karlsson, Oct 29 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A322577(n/d) * a(d).

a(n) = A347093(n) - A322577(n).

From Sebastian Karlsson, Oct 29 2021: (Start)

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

a(n) = Sum_{d|n} A323363(n/d)*A023900(d).

Multiplicative with a(p^e) = 1 + p^2 if e is even, -2*p if e is odd. (End)