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Number of divisors of 2n.
30

%I #52 May 11 2024 11:02:34

%S 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 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 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 Number of divisors of 2n.

%H Antti Karttunen, <a href="/A099777/b099777.txt">Table of n, a(n) for n = 1..65537</a>

%F Moebius transform is period 2 sequence [2, 1, ...]. - _Michael Somos_, Sep 20 2005

%F 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 a(n) = A000005(n) + A001227(n). - _Matthew Vandermast_, Sep 30 2014

%F 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 From _Bernard Schott_, Sep 14 2020: (Start)

%F a(n) = 2 iff n = 1;

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

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

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

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

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

%t DivisorSigma[0, 2*Range[100]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 20 2011 *)

%o (PARI) a(n)=if(n<1, 0, numdiv(2*n)) /* _Michael Somos_, Sep 20 2005 */

%Y Bisection of A000005.

%Y Cf. A099774, A001248, A065091, A100484, A372713.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Nov 19 2004

%E More terms from _Emeric Deutsch_, Dec 03 2004