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A355448
a(n) = 1 if the number of divisors of n^2 is coprime to 6, otherwise 0.
3
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1
OFFSET
1
COMMENTS
According to the formula in A048691, n is coprime to 6 if the exponents in the canonical prime factorization of n are a subset of A007494, i.e., none of the exponents is in A016777. An immediate consequence is that a(n)=|A307424(n)| is true. - R. J. Mathar, Feb 07 2023
FORMULA
Multiplicative with a(p^e) = [gcd(2*e+1,6) == 1], where [ ] is the Iverson bracket.
a(n) = A354354(A048691(n)).
a(n) = abs(A307424(n)). [conjectured]
a(n) = Sum_{d|n} A010057(n/d) * A227291(d), Dirichlet convolution of A010057 by A227291.
Dirichlet g.f.: zeta(2*s) * zeta(3*s) / zeta(4*s). - Vaclav Kotesovec, Jul 15 2022
Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = zeta(3/2)/zeta(2) = 1.5881337746... . - Amiram Eldar, Oct 05 2023
MATHEMATICA
Table[If[CoprimeQ[DivisorSigma[0, n^2], 6], 1, 0], {n, 1, 100}] (* Vaclav Kotesovec, Jul 14 2022 *)
PROG
(PARI) A355448(n) = (1==gcd(numdiv(n^2), 6));
(PARI) A355448(n) = factorback(apply(e->(1==gcd(e+e+1, 6)), factor(n)[, 2]));
CROSSREFS
Characteristic function of A350014.
Absolute values of A307424. [conjectured]
Cf. A000005, A010057, A048691, A227291, A307421, A354354, A355684 (Dirichlet inverse).
Cf. also A353470.
Sequence in context: A170956 A293449 A307424 * A354868 A075802 A112526
KEYWORD
nonn,easy,mult
AUTHOR
Antti Karttunen, Jul 13 2022
STATUS
approved