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

1,1

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

Formula

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

For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353457(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)).

Prog

(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(d<n, A353457(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

Crossrefs

Cf. A003961, A348717, A353457, A353458.

Cf. also A353469.

Sequence in context: A307303 A324252 A321445 * A007423 A076544 A349912

Adjacent sequences: A353456 A353457 A353458 * A353460 A353461 A353462

Keyword

sign

Author

Antti Karttunen, Apr 21 2022

Status

approved