A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.

1, -1, -1, 4, -1, 1, -1, -4, 9, 1, -1, -4, -1, 1, 1, 16, -1, -9, -1, -4, 1, 1, -1, 4, 25, 1, -9, -4, -1, -1, -1, -16, 1, 1, 1, 36, -1, 1, 1, 4, -1, -1, -1, -4, -9, 1, -1, -16, 49, -25, 1, -4, -1, 9, 1, 4, 1, 1, -1, 4, -1, 1, -9, 64, 1, -1, -1, -4, 1, -1, -1, -36, -1, 1, -25, -4, 1, -1, -1, -16

Offset: 1

Author

Werner Schulte, Nov 07 2022

Keywords

• sign
• easy
• mult

Comments

Signed version of A008833.

Crossrefs

Cf. A000010, A008833, A008836, A034444, A061019.

Programs

Maple

A358272 := proc(n)
    local a,pe,e,p ;
    a := 1;
    for pe in ifactors(n)[2] do
        e := op(2,pe) ;
        p := op(1,pe) ;
        a := a*(-1)^e*p^(2*floor(e/2)) ;
    end do:
    a ;
end proc:
seq(A358272(n),n=1..80) ; # R. J. Mathar, Jan 17 2023

Mathematica

f[p_, e_] := (-1)^e * p^(2*Floor[e/2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)

Python

from math import prod
from sympy import factorint
def A358272(n): return prod(-p**(e&-2) if e&1 else p**(e&-2) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 17 2023

Formula

a(n) = lambda(n) * A008833(n) for n > 0 where lambda(n) = A008836(n).

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

Dirichlet inverse b(n), n > 0, is multiplicative with b(p) = 1 and b(p^e) = 1 - p^2 for prime p and e > 1.

Dirichlet convolution with A034444 equals A008833.

Equals Dirichlet convolution of A000010 and A061019.

Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * lambda(gcd(k, n)) for n > 0.

a(n) = Sum_{d|n} lambda(d)*d*phi(n/d), where lambda(n) = A008836(n). - Ridouane Oudra, May 11 2025