A074823 a(n) = 2^omega(n)*mu(n)^2.

1, 2, 2, 0, 2, 4, 2, 0, 0, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0, 4, 4, 2, 0, 0, 4, 0, 0, 2, 8, 2, 0, 4, 4, 4, 0, 2, 4, 4, 0, 2, 8, 2, 0, 0, 4, 2, 0, 0, 0, 4, 0, 2, 0, 4, 0, 4, 4, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 4, 8, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 0, 4, 2, 0, 4, 4, 4, 0, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 0, 2, 8, 2, 0, 8

Offset: 1

Author

Benoit Cloitre, Sep 08 2002

Keywords

• mult
• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A001221, A008683, A008966, A034444, A069201, A226177, A347149.

Programs

Mathematica

Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 100}] (* Vaclav Kotesovec, Aug 20 2021 *)
f[p_, e_] :=If[e==1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

PARI

a(n) = 2^omega(n)*moebius(n)^2; \\ Michel Marcus, Jul 23 2017

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

Scheme

(define (A074823 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 0) (A074823 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

Formula

Sum_{k=1..n} a(k) = A069201(n).

Multiplicative with a(p)=2, a(p^e)=0, e > 1.

a(n) = A034444(n)*A008966(n). - R. J. Mathar, Apr 15 2011

Sum_{n>0} a(n)/n^s = Product_{p prime} (1 + 2p^(-s)). - Ralf Stephan, Jul 07 2013

a(n) = abs(A226177(n)). - Antti Karttunen, Jul 23 2017

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

Extensions

Additional comments from Vladeta Jovovic, Dec 30 2002