A355448 a(n) = 1 if the number of divisors of n^2 is coprime to 6, otherwise 0.

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

Links

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

R. J. Mathar, OEIS A355448.

Index entries for characteristic functions.

Index entries for sequences computed from exponents in factorization of n.

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.

Cf. A013661, A078434.

Sequence in context: A170956 A293449 A307424 * A354868 A075802 A112526

Adjacent sequences: A355445 A355446 A355447 * A355449 A355450 A355451

Keyword

nonn,easy,mult

Author

Antti Karttunen, Jul 13 2022

Status

approved