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
Examples
a(7) = 4 because the divisors of 14 are: 1, 2, 7 and 14.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
Programs
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Maple
with(numtheory): seq(tau(2*n),n=1..100);
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Mathematica
DivisorSigma[0, 2*Range[100]] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)
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PARI
a(n)=if(n<1, 0, numdiv(2*n)) /* Michael Somos, Sep 20 2005 */
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
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)
Dirichlet g.f.: zeta(s)^2 * (2 - 1/2^s). - Amiram Eldar, Jun 08 2025
Extensions
More terms from Emeric Deutsch, Dec 03 2004
Comments