A271102 a(n) is multiplicative with a(p^e) = -1 if e=2, a(p^e) = 0 if e=1 or e>2.

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

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

Formula

Dirichlet g.f.: 1 / zeta(2*s).

abs(a(n)) = A227291(n).

Moebius transform of A008966. A008966(n) = abs(mu(n)) = Sum_{d|n} a(d), where mu is the Moebius function (A008683). - Michael Somos, Jan 30 2020

a(n) = Sum_{d|n} mu(d)*mu(n/d)^2. - Ridouane Oudra, Jan 21 2021

Example

G.f. = x - x^4 - x^9 - x^25 + x^36 - x^49 + x^100 - x^121 - x^169 + ...

Mathematica

Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> If[e == 2, -1, 0]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)
Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, 100}] (* Vaclav Kotesovec, Apr 08 2019 *)

Prog

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 - X^2 )[n])};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; -(e==2)))};
(Scheme) (define (A271102 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) -1 0) (A271102 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

Crossrefs

Cf. A008966, A227291 (gives the absolute values), Dirichlet inverse of A010052.

Sequence in context: A014504 A014999 A227291 * A353800 A353374 A326072

Adjacent sequences: A271099 A271100 A271101 * A271103 A271104 A271105

Keyword

sign,mult

Author

Michael Somos, Mar 30 2016

Extensions

More terms from Antti Karttunen, Jul 28 2017

Status

approved