A099777 Number of divisors of 2n.

2, 3, 4, 4, 4, 6, 4, 5, 6, 6, 4, 8, 4, 6, 8, 6, 4, 9, 4, 8, 8, 6, 4, 10, 6, 6, 8, 8, 4, 12, 4, 7, 8, 6, 8, 12, 4, 6, 8, 10, 4, 12, 4, 8, 12, 6, 4, 12, 6, 9, 8, 8, 4, 12, 8, 10, 8, 6, 4, 16, 4, 6, 12, 8, 8, 12, 4, 8, 8, 12, 4, 15, 4, 6, 12, 8, 8, 12, 4, 12, 10, 6, 4, 16, 8, 6, 8, 10, 4, 18, 8, 8, 8, 6, 8

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

Moebius transform is period 2 sequence [2, 1, ...]. - Michael Somos, Sep 20 2005

G.f.: Sum_{k>0} x^k(2+x^k)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1))+x^(2k)/(1-x^(2k)). - Michael Somos, Sep 20 2005

a(n) = A000005(n) + A001227(n). - Matthew Vandermast, Sep 30 2014

Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

From Bernard Schott, Sep 14 2020: (Start)

a(n) = 2 iff n = 1;

a(n) = prime(m) iff n = 2^(prime(m)-2);

a(n) = 4 iff n = 4 or n is an odd prime (A065091);

a(n) = 6 iff n = 16, or n = 2p for p an odd prime (A100484 \ {4}), or n = p^2 for p an odd prime (A001248 \ {4});

a(n) = 2*A000005(n) iff n is odd. (End)

Example

a(7)=4 because the divisors of 14 are: 1,2,7 and 14.

Maple

with(numtheory): seq(tau(2*n), n=1..100);

Mathematica

DivisorSigma[0, 2*Range[100]] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

Prog

(PARI) a(n)=if(n<1, 0, numdiv(2*n)) /* Michael Somos, Sep 20 2005 */

Crossrefs

Bisection of A000005.

Cf. A099774, A001248, A065091, A100484.

Sequence in context: A087875 A195848 A342731 * A221917 A131798 A206925

Adjacent sequences: A099774 A099775 A099776 * A099778 A099779 A099780

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 19 2004

Extensions

More terms from Emeric Deutsch, Dec 03 2004

Status

approved