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 A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions. 23
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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 Also number of decompositions of divisors of n into coprime pairs. - K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001 Number of elements in the set {(x,y): x|n, y|n, x 2, then a(n) > A062799(n). A001221(n), or omega(n), is the number of distinct primes dividing 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 side length. Equivalent to the even indices of A046081. - Nathaniel C Beckman, May 14 2020 REFERENCES 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. 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 443-4 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. 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... Bisection of A046079. - Lekraj Beedassy, Jul 09 2004 a(n) = Sum_{i=1..2*n-1} chi(i*(4*n-i)/(4*n-2*i)), where chi is the integer characteristic. - Wesley Ivan Hurt, Apr 24 2020 EXAMPLE 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). MATHEMATICA Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}] (DivisorSigma[0, Range[100]^2]-1)/2 (* Harvey P. Dale, Apr 15 2013 *) PROG (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 (MAGMA) [(NumberOfDivisors(n^2)-1)/2 : n in [1..100]]; // Vincenzo Librandi, Apr 18 2018 CROSSREFS Cf. A018892, A063427, A063428. First twenty-nine terms identical to those of A062799. Cf. A063717, A063718, A048691. Sequence in context: A067614 A113901 A062799 * A263653 A330328 A269427 Adjacent sequences:  A063644 A063645 A063646 * A063648 A063649 A063650 KEYWORD nonn,easy,nice,changed AUTHOR Henry Bottomley, Jul 23 2001 STATUS approved

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Last modified May 29 03:06 EDT 2020. Contains 334696 sequences. (Running on oeis4.)