A162512 Dirichlet inverse of A162511.

1, -1, -1, 2, -1, 1, -1, -4, 2, 1, -1, -2, -1, 1, 1, 8, -1, -2, -1, -2, 1, 1, -1, 4, 2, 1, -4, -2, -1, -1, -1, -16, 1, 1, 1, 4, -1, 1, 1, 4, -1, -1, -1, -2, -2, 1, -1, -8, 2, -2, 1, -2, -1, 4, 1, 4, 1, 1, -1, 2, -1, 1, -2, 32, 1, -1, -1, -2, 1, -1, -1, -8, -1, 1, -2, -2, 1, -1, -1, -8, 8, 1

Offset: 1

Author

Gerard P. Michon, Jul 05 2009, Jul 06 2009

Keywords

• easy
• mult
• sign

Comments

The absolute value of this sequence is A162510.

The Moebius function (A008683) can be defined in terms of this sequence: A008683(n) is equal to a(n) if a(n) is odd and zero otherwise.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• G. P. Michon, Multiplicative functions.

Crossrefs

Cf. A005117, A008683, A067029, A076479, A162510, A162511.

Programs

Maple

A162512 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := -a*(-2)^(op(2,f)-1) ;
    end do:
    return a;
end proc:
seq(A162512(n),n=1..100) ; # R. J. Mathar, May 20 2017

Mathematica

b[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]);
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

PARI

a(n) = my(f=factor(n)); for(i=1, #f~, f[i,1]=-(-2)^(f[i,2]-1); f[i,2]=1); factorback(f); \\ Michel Marcus, May 20 2017

Python

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [-(-2)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

Scheme

(define (A162512 n) (if (= 1 n) n (* (- (expt -2 (- (A067029 n) 1))) (A162512 (A028234 n))))) ;; Antti Karttunen, May 20 2017, after the given multiplicative formula.

Formula

Multiplicative function with a(p^e)=-(-2)^(e-1) for any prime p and any positive exponent e.

a(n) = n/2 when n is a power of 4 (A000302).

a(n) = A008683(n) when n is a squarefree number (A005117).

Dirichlet g.f.: Product_{p prime} ((p^s + 1)/(p^s + 2)). - Amiram Eldar, Oct 26 2023