cp's OEIS Frontend

A099777 Number of divisors of 2n.

%I A099777 #57 Jun 08 2025 02:41:26
%S A099777 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,
%T A099777 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,
%U A099777 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
%N A099777 Number of divisors of 2n.
%H A099777 Antti Karttunen, <a href="/A099777/b099777.txt">Table of n, a(n) for n = 1..65537</a>
%F A099777 Moebius transform is period 2 sequence [2, 1, ...]. - _Michael Somos_, Sep 20 2005
%F A099777 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
%F A099777 a(n) = A000005(n) + A001227(n). - _Matthew Vandermast_, Sep 30 2014
%F A099777 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
%F A099777 From _Bernard Schott_, Sep 14 2020: (Start)
%F A099777 a(n) = 2 iff n = 1;
%F A099777 a(n) = prime(m) iff n = 2^(prime(m)-2);
%F A099777 a(n) = 4 iff n = 4 or n is an odd prime (A065091);
%F A099777 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});
%F A099777 a(n) = 2*A000005(n) iff n is odd. (End)
%F A099777 Dirichlet g.f.: zeta(s)^2 * (2 - 1/2^s). - _Amiram Eldar_, Jun 08 2025
%e A099777 a(7) = 4 because the divisors of 14 are: 1, 2, 7 and 14.
%p A099777 with(numtheory): seq(tau(2*n),n=1..100);
%t A099777 DivisorSigma[0, 2*Range[100]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 20 2011 *)
%o A099777 (PARI) a(n)=if(n<1, 0, numdiv(2*n)) /* _Michael Somos_, Sep 20 2005 */
%Y A099777 Bisection of A000005.
%Y A099777 Cf. A099774, A001248, A065091, A100484, A372713.
%K A099777 nonn,easy
%O A099777 1,1
%A A099777 _N. J. A. Sloane_, Nov 19 2004
%E A099777 More terms from _Emeric Deutsch_, Dec 03 2004