This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066660 #52 Dec 10 2024 11:58:56
%S A066660 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 A066660 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 A066660 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 A066660 Number of divisors of 2n excluding 1.
%C A066660 a(n) is the number of integers of the form (n+k)/(n-k) for k=0,1,2,...,n-1.
%C A066660 Inverse Moebius transform of A040001 (offset 1).
%C A066660 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 A066660 Harry J. Smith, <a href="/A066660/b066660.txt">Table of n, a(n) for n = 1..1000</a>
%H A066660 Vaclav Kotesovec, <a href="/A066660/a066660.jpg">Graph - the asymptotic ratio</a>
%F A066660 a(n) = A069930(n) + 1.
%F A066660 If n is an odd prime, then a(n)=3.
%F A066660 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 A066660 Also lim_{n -> infinity} card(i<n, a(i) even)/card(i<n, a(i) odd) = 0.
%F A066660 G.f.: Sum_{n>0} x^n(1 - x^(3n))/((1 - x^n)(1 - x^(2n))).
%F A066660 a(n) = d(2n) - 1, where d(n) is the number of divisors of n (A000005). - _Wesley Ivan Hurt_, Dec 22 2013
%F A066660 a(n) = n - A234306(n). - _Antti Karttunen_, Dec 22 2013
%F A066660 a(n) = Sum_{i=1..n} floor(2*n/i) - floor((2*n-1)/i). - _Wesley Ivan Hurt_, Nov 15 2017
%F A066660 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 A066660 a(4)=3 because (4+0)/(4-0), (4+2)/(4-2), (4+3)/(4-3) are integers.
%p A066660 with(numtheory); A066660:=n->tau(2*n)-1; seq(A066660(n), n=1..100); # _Wesley Ivan Hurt_, Dec 22 2013
%t A066660 Table[DivisorSigma[0, 2 n] - 1, {n, 100}] (* _Wesley Ivan Hurt_, Dec 22 2013 *)
%o A066660 (PARI) a(n)=if(n<1,0,sumdiv(n,d,(d>1)+d%2))
%o A066660 (PARI) {a(n)=if(n<1, 0, numdiv(2*n)-1)} /* _Michael Somos_, Sep 03 2006 */
%o A066660 (Magma) [DivisorSigma(0,2*n) -1: n in [1..100]]; // _G. C. Greubel_, Feb 13 2019
%o A066660 (Sage) [sigma(2*n,0) -1 for n in (1..100)] # _G. C. Greubel_, Feb 13 2019
%Y A066660 Cf. A000005, A040001, A234306.
%K A066660 nonn,easy
%O A066660 1,2
%A A066660 _Benoit Cloitre_, Jan 11 2002