A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 13, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 13, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 22, 1, 4, 7, 6, 4, 13, 1, 7, 4, 13, 1, 17, 1, 4, 7, 7, 4, 13, 1, 13, 4, 4, 1, 22, 4, 4, 4, 10, 1, 22, 4, 7, 4, 4, 4

Offset: 1

Author

Henry Bottomley, Jul 23 2001

Keywords

• nonn
• easy
• nice

Comments

Also number of ways to write 1/n as sum of exactly two distinct unit fractions. - Thomas L. York, Jan 11 2014

Also number of positive integers m such that 1/n + 1/m is a unit fraction. - Jon E. Schoenfield, Apr 17 2018

If 1/n = 1/b - 1/c then n = bc/(c-b) and 1/n = 1/(2n-b) + 1/(c+2n) (though it is also the case that 1/n = 1/(2n) + 1/(2n) equivalent to b = c = 0).

Also number of divisors of n^2 less than n. - Vladeta Jovovic, Aug 13 2001

Number of elements in the set {(x,y): x|n, y|n, xVladeta Jovovic, May 03 2002

Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre, Jan 04 2002

This is similar to A062799, having the same first 29 terms. But they are different sequences.

If A001221(n) = omega(n) <= 2, then a(n) = A062799(n); if A001221(n) > 2, then a(n) > A062799(n). - Matthew Vandermast, Aug 25 2004

Number of r X s integer-sided rectangles such that r + s = 4n, r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020

Also number of integer-sided right triangles with 2n as a leg. Equivalent to the even indices of A046079. - Nathaniel C Beckman, May 14 2020; Jun 26 2020

a(n) is the number of positive integers k such that k+n divides k*n. - Thomas Ordowski, Dec 02 2024

Examples

			a(10) = 4 since 1/10 = 1/5 - 1/10 = 1/6 - 1/15 = 1/8 - 1/40 = 1/9 - 1/90.
a(12) = 7: the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the decompositions are (1, 2), (1, 3), (1, 4), (1, 6), (1, 12), (2, 3), (3, 4).
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Christopher J. Bradley, Solution to Problem 2175, Crux Mathematicorum, Vol. 23, No. 7, (Nov 1997), pp. 443-444.
• Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
• Roger B. Eggleton, Unitary Fractions: 10501, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 372.
• Amiram Eldar, Plot of (Sum_{i=1..n} a(i))/f(n) - 1, where f(n) is an asymptotic function, for n = 2^(1..28).
• Amiram Eldar, Plot of Sum_{i=1..n} a(i)/(n*log(n)*log(log(n))) for n = 2^(1..28).
• Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001, pp. 303-306.

Crossrefs

Cf. A018892, A063427, A063428, A129296.

First twenty-nine terms identical to those of A062799.

Cf. A001221, A046079, A048691, A063717, A063718.

Cf. A001620, A082633, A013661, A201994, A306016.

Programs

Magma

[(NumberOfDivisors(n^2)-1)/2 : n in [1..100]]; // Vincenzo Librandi, Apr 18 2018

Mathematica

Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
(DivisorSigma[0,Range[100]^2]-1)/2 (* Harvey P. Dale, Apr 15 2013 *)

PARI

for(n=1,100,print1(sum(i=1,n^2,if((n*i)%(i+n),0,1)),","))

PARI

a(n)=numdiv(n^2)\2 \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = (tau(n^2)-1)/2.

a(n) = A018892(n)-1. If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1+1)(2*a2+1)...(2*at+1)-1)/2.

If n is prime a(n)=1. Conjecture: (1/n)*Sum_{i=1..n} a(i) = C*log(n)*log(log(n)) + o(log(n)) with C=0.7... [The conjecture is false. See the plot and the asymptotic formula below. - Amiram Eldar, Oct 03 2024]

Bisection of A046079. - Lekraj Beedassy, Jul 09 2004

a(n) = Sum_{i=1..2*n-1} (1 - ceiling(i*(4*n-i)/(4*n-2*i)) + floor(i*(4*n-i)/(4*n-2*i))). - Wesley Ivan Hurt, Apr 24 2020

Sum_{k=1..n} a(k) ~ (n/(2*zeta(2)))*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 - zeta(2) + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Oct 03 2024