A349343 Dirichlet inverse of A193356, which is defined as n if n is odd, 0 if n is even.

1, 0, -3, 0, -5, 0, -7, 0, 0, 0, -11, 0, -13, 0, 15, 0, -17, 0, -19, 0, 21, 0, -23, 0, 0, 0, 0, 0, -29, 0, -31, 0, 33, 0, 35, 0, -37, 0, 39, 0, -41, 0, -43, 0, 0, 0, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 0, -59, 0, -61, 0, 0, 0, 65, 0, -67, 0, 69, 0, -71, 0, -73, 0, 0, 0, 77, 0, -79, 0, 0, 0, -83, 0, 85, 0, 87, 0, -89

Offset: 1

Author

Antti Karttunen, Nov 15 2021

Keywords

• sign
• easy
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Agrees with A349341 on odd numbers.

Cf. A193356, and also A328203, A349134, A349344, A349346, A349353.

Programs

Mathematica

a[1]=1;a[n_]:=-DivisorSum[n,If[OddQ[n/#],n/#,0]*a@#&,#Giorgos Kalogeropoulos, Nov 15 2021 *)
f[p_, e_] := If[e == 1, -p, 0]; f[2, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

PARI

A349343(n) = { my(f = factor(n)); prod(i=1, #f~, if((2==f[i,1])||(f[i,2]>1), 0, -f[i,1])); };

Formula

a(2n) = 0, a(2n+1) = A349341(2n+1) for all n >= 1.

Multiplicative with a(p^e) = 0 if p=2 or e>1, otherwise a(p) = -p. - (After Sebastian Karlsson's similar formula for A349341).