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A069930 Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1. 4
0, 1, 2, 2, 2, 4, 2, 3, 4, 4, 2, 6, 2, 4, 6, 4, 2, 7, 2, 6, 6, 4, 2, 8, 4, 4, 6, 6, 2, 10, 2, 5, 6, 4, 6, 10, 2, 4, 6, 8, 2, 10, 2, 6, 10, 4, 2, 10, 4, 7, 6, 6, 2, 10, 6, 8, 6, 4, 2, 14, 2, 4, 10, 6, 6, 10, 2, 6, 6, 10, 2, 13, 2, 4, 10, 6, 6, 10, 2, 10, 8, 4, 2, 14, 6, 4, 6, 8, 2, 16, 6, 6, 6, 4, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Number of r X s integer-sided rectangles such that r < s, r + s = 2n and r | s. - Wesley Ivan Hurt, Apr 24 2020

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A032741(n) + A069283(n) = A000005(n) - 1 + A001227(n) - 1 = tau(n) + A001227(n) - 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002

Asymptotic formula: since sum(k=1, n, a(k)) = sum(k=1, n, tau(k)) + sum(k=1, n, A001227(k)) - 2*n = A006218(n) + A060831(n) - 2*n = 2*A006218(n) - A006218(floor(n/2)) - 2*n with A006218(0) = 0, A006218(n) = sum(k=1, n, tau(k)) and now, by Dirichlet's asymptotic expression A006218(n) = n*log(n) + n*(2*gamma-1) + O(n^theta) (gamma = 0.57721..; 1/4 <= theta < 1/2), we have sum(k=1, n, a(k)) = 2*n*log(n) - (n/2)*log(n) + o(n*log(n)) = 1.5*n*log(n) + o(n*log(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 13 2002

a(n) = tau(2*n) - 2. - Michael Somos, Aug 30 2012

Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 7), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019

MATHEMATICA

Table[Count[Table[(n+k)/(n-k), {k, n-1}], _?IntegerQ], {n, 100}] (* Harvey P. Dale, Jun 04 2019 *)

PROG

(PARI) for(n=1, 150, print1(sum(i=1, n-1, if((n+i)%(n-i), 0, 1)), ", "))

(PARI) {a(n) = if( n<1, 0, numdiv(2*n) - 2)} /* Michael Somos, Aug 30 2012 */

CROSSREFS

Cf. A032741, A069283, A000005, A001227, A006218, A060831, A066743.

A066660(n) - 1.

Sequence in context: A029658 A332889 A239452 * A086327 A114896 A216620

Adjacent sequences:  A069927 A069928 A069929 * A069931 A069932 A069933

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 05 2002

STATUS

approved

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Last modified October 24 16:24 EDT 2021. Contains 348233 sequences. (Running on oeis4.)