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A066660 Number of divisors of 2n excluding 1. 5
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Inverse Moebius transform of A040001 (offset 1).

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

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000

Vaclav Kotesovec, Graph - the asymptotic ratio

FORMULA

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

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

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]

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

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

a(n) = d(2n) - 1, where d(n) is the number of divisors of n (A000005). - Wesley Ivan Hurt, Dec 22 2013

a(n) = n - A234306(n). - Antti Karttunen, Dec 22 2013

a(n) = Sum_{i=1..n} floor(2*n/i) - floor((2*n-1)/i). - Wesley Ivan Hurt, Nov 15 2017

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

Table[DivisorSigma[0, 2 n] - 1, {n, 100}] (* Wesley Ivan Hurt, Dec 22 2013 *)

PROG

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

(PARI) {a(n)=if(n<1, 0, numdiv(2*n)-1)} /* Michael Somos, Sep 03 2006 */

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

(MAGMA) [DivisorSigma(0, 2*n) -1: n in [1..100]]; // G. C. Greubel, Feb 13 2019

(Sage) [sigma(2*n, 0) -1 for n in (1..100)] # G. C. Greubel, Feb 13 2019

CROSSREFS

Cf. A000005, A040001, A234306.

Sequence in context: A197592 A103359 A020481 * A057957 A241686 A076559

Adjacent sequences:  A066657 A066658 A066659 * A066661 A066662 A066663

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Jan 11 2002

STATUS

approved

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Last modified March 31 16:23 EDT 2020. Contains 333151 sequences. (Running on oeis4.)