A359166 a(n) = lambda(n) * lambda(sigma(n)), where lambda is Liouville's lambda, and sigma is the sum of divisors function.

1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, 1

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences related to sigma(n)

Formula

Multiplicative with a(p^e) = (-1)^(e + A001222(1 + p + p^2 + ... + p^e)).

a(n) = A008836(n) * A358766(n) = A008836(n) * A008836(A000203(n)).

Prog

(PARI) A359166(n) = ((-1)^(bigomega(n)+bigomega(sigma(n))));
(Python)
from functools import reduce
from operator import ixor
from collections import Counter
from sympy import factorint
def A359166(n): return (-1 if reduce(ixor, (f:=factorint(n)).values(), 0)&1 else 1)*(-1 if reduce(ixor, sum((Counter(factorint((p**(e+1)-1)//(p-1))) for p, e in f.items()), Counter()).values(), 0)&1 else 1) # Chai Wah Wu, Dec 23 2022

Crossrefs

Cf. A000203, A001222, A008836, A058063, A358766, A359167 (positions of positive terms), A359168 (of negative terms).

Sequence in context: A186035 A358766 A359154 * A359581 A359792 A362927

Adjacent sequences: A359163 A359164 A359165 * A359167 A359168 A359169

Keyword

sign,mult

Author

Antti Karttunen, Dec 19 2022

Status

approved