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

Author

N. J. A. Sloane, Nov 20 2004

Keywords

• nonn
• easy

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

Examples

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

Links

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

Crossrefs

Bisection of A034444, twice A068068.

Cf. A000079, A001221, A099812.

Programs

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 *)

PARI

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

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

Extensions

More terms from Emeric Deutsch, Dec 24 2004