A353459 Sum of A353457 and its Dirichlet inverse.

2, 0, 0, 1, 0, -2, 0, 1, 1, 2, 0, -1, 0, -2, -2, 1, 0, -1, 0, 1, 2, 2, 0, -1, 1, -2, 1, -1, 0, 0, 0, 1, -2, 2, -2, 0, 0, -2, 2, 1, 0, 0, 0, 1, -1, 2, 0, -1, 1, 1, -2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 0, 2, 1, 1, -2, 0, 0, 1, -2, 0, 0, 0, 0, -2, -1, -1, -2, 0, 0, 1, 1, 2, 0, 1, 2, -2, 2, 1, 0, 1, 2, 1, -2, 2, -2, -1, 0

Offset: 1

Author

Antti Karttunen, Apr 21 2022

Keywords

• sign

Comments

Only values in range {-2, -1, 0, +1, +2} occur.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A003961, A348717, A353457, A353458.

Cf. also A353469.

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA353457 = Map();
A353457(n) = if(1==n,1,my(v); if(mapisdefined(memoA353457,n,&v), v, v = -sumdiv(n,d,if(dA353457(A064989(n/d))*A353457(d),0)); mapput(memoA353457,n,v); (v)));
A353459(n) = (A353457(n)+A353457(A064989(n)));

Python

from math import prod
from sympy import factorint, primepi
def A353459(n):
    f = [(primepi(p)&1, -int(e==1)) for p, e in factorint(n).items()]
    return prod(e for p, e in f if not p)+prod(e for p, e in f if p) # Chai Wah Wu, Jan 05 2023

Formula

a(n) = A353457(n) + A353458(n) = A353457(n) + A353457(A064989(n)).

For n > 1, a(n) = -Sum_{d|n, 1A353457(d) * A353458(n/d). [As the sequences are Dirichlet inverses of each other]

For all n >= 1, a(n) = a(A003961(n)) = a(A348717(n)).