A100008 Number of unitary divisors of 2n.

2, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 4, 8, 2, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 4, 4, 8, 4, 2, 8, 4, 8, 4, 4, 4, 8, 4, 4, 8, 4, 4, 8, 4, 4, 4, 4, 4, 8, 4, 4, 4, 8, 4, 8, 4, 4, 8, 4, 4, 8, 2, 8, 8, 4, 4, 8, 8, 4, 4, 4, 4, 8, 4, 8, 8, 4, 4, 4, 4, 4, 8, 8, 4, 8, 4, 4, 8, 8, 4, 8, 4, 8, 4, 4, 4, 8, 4, 4, 8, 4, 4, 16

Offset

1,1

Comments

b(n) = a(n)/a(1) is multiplicative with b(2^e) = 1, b(p^e) = 2 otherwise. - David W. Wilson, Jun 12 2005

Links

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

Formula

a(n) = A000079(A099812(n)) = A000079(A001221(2n)) = 2*A068068(n). - Antti Karttunen, Sep 14 2017

Dirichlet g.f.: 2*zeta(s)^2/(zeta(2*s)*(1+1/2^s)). - Amiram Eldar, Jan 28 2023

Sum_{k=1..n} a(k) ~ 8*n*((log(n) - 1 + 2*gamma + log(2)/3)/Pi^2 - 12*zeta'(2)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 28 2023

Example

a(6)=4 because among the six divisors of 12 only 1,3,4 and 12 are unitary.

Maple

with(numtheory): for n from 1 to 120 do printf(`%d, `, 2^nops(ifactors(2*n)[2])) od: # Emeric Deutsch, Dec 24 2004

Mathematica

a[n_] := 2^PrimeNu[2*n]; Array[a, 100] (* Amiram Eldar, Jan 28 2023 *)

Prog

(PARI) A100008(n) = 2^omega(2*n); \\ Antti Karttunen, Sep 14 2017

Crossrefs

Bisection of A034444, twice A068068.

Cf. A000079, A001221, A099812.

Sequence in context: A091248 A082991 A214212 * A102763 A054844 A057936

Adjacent sequences: A100005 A100006 A100007 * A100009 A100010 A100011

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 20 2004

Extensions

More terms from Emeric Deutsch, Dec 24 2004

Status

approved