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A069930
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Number of integers of the form (n+k)/(n-k) with 1 <= k <= n-1.
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5
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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
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OFFSET
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1,3
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COMMENTS
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Number of r X s integer-sided rectangles such that r < s, r + s = 2n and r | s. - Wesley Ivan Hurt, Apr 24 2020
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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Table[Count[Table[(n+k)/(n-k), {k, n-1}], _?IntegerQ], {n, 100}] (* Harvey P. Dale, Jun 04 2019 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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