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Number of divisors of 2n excluding 1.
6

%I #48 Sep 08 2022 08:45:04

%S 1,2,3,3,3,5,3,4,5,5,3,7,3,5,7,5,3,8,3,7,7,5,3,9,5,5,7,7,3,11,3,6,7,5,

%T 7,11,3,5,7,9,3,11,3,7,11,5,3,11,5,8,7,7,3,11,7,9,7,5,3,15,3,5,11,7,7,

%U 11,3,7,7,11,3,14,3,5,11,7,7,11,3,11,9,5,3,15,7,5,7,9,3,17,7,7,7,5,7

%N Number of divisors of 2n excluding 1.

%C a(n) is the number of integers of the form (n+k)/(n-k) for k=0,1,2,...,n-1.

%C Inverse Moebius transform of A040001 (offset 1).

%C The number of partitions of 2n into exactly two parts (2n-i,i) such that i divides (2n-i). - _Wesley Ivan Hurt_, Dec 22 2013

%H Harry J. Smith, <a href="/A066660/b066660.txt">Table of n, a(n) for n = 1..1000</a>

%H Vaclav Kotesovec, <a href="/A066660/a066660.jpg">Graph - the asymptotic ratio</a>

%F a(n) = A069930(n) + 1.

%F If n is an odd prime, then a(n)=3.

%F Asymptotic formula: 1/n*Sum(i=1, n, a(i)) = C*log(n) + o(log(n)) with C=3/2. [corrected by _Vaclav Kotesovec_, Feb 13 2019]

%F Also lim_{n -> infinity} card(i<n, a(i) even)/card(i<n, a(i) odd) = 0.

%F G.f.: Sum_{n>0} x^n(1 - x^(3n))/((1 - x^n)(1 - x^(2n))).

%F a(n) = d(2n) - 1, where d(n) is the number of divisors of n (A000005). - _Wesley Ivan Hurt_, Dec 22 2013

%F a(n) = n - A234306(n). - _Antti Karttunen_, Dec 22 2013

%F a(n) = Sum_{i=1..n} floor(2*n/i) - floor((2*n-1)/i). - _Wesley Ivan Hurt_, Nov 15 2017

%F Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 5), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 13 2019

%e a(4)=3 because (4+0)/(4-0), (4+2)/(4-2), (4+3)/(4-3) are integers.

%p with(numtheory); A066660:=n->tau(2*n)-1; seq(A066660(n), n=1..100); # _Wesley Ivan Hurt_, Dec 22 2013

%t Table[DivisorSigma[0, 2 n] - 1, {n, 100}] (* _Wesley Ivan Hurt_, Dec 22 2013 *)

%o (PARI) a(n)=if(n<1,0,sumdiv(n,d,(d>1)+d%2))

%o (PARI) {a(n)=if(n<1, 0, numdiv(2*n)-1)} /* _Michael Somos_, Sep 03 2006 */

%o (PARI) { for (n=1, 1000, write("b066660.txt", n, " ", numdiv(2*n) - 1) ) } \\ _Harry J. Smith_, Mar 16 2010

%o (Magma) [DivisorSigma(0,2*n) -1: n in [1..100]]; // _G. C. Greubel_, Feb 13 2019

%o (Sage) [sigma(2*n,0) -1 for n in (1..100)] # _G. C. Greubel_, Feb 13 2019

%Y Cf. A000005, A040001, A234306.

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Jan 11 2002