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A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions. 21
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

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<y, GCD(x,y)=1}. - Vladeta 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 sequence.

If A001221(n)<=2, then a(n)=A062799(n); if A001221(n)>2, then a(n)>A062799(n). A001221(n), or omega(n), is the number of distinct primes dividing n. - Matthew Vandermast, Aug 25 2004

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

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

CROSSREFS

Cf. A018892, A063427, A063428. First twenty-nine terms identical to those of A062799 (offset).

Cf. A063717, A063718, A048691.

Sequence in context: A067614 A113901 A062799 * A263653 A269427 A077808

Adjacent sequences:  A063644 A063645 A063646 * A063648 A063649 A063650

KEYWORD

nonn,easy,nice

AUTHOR

Henry Bottomley, Jul 23 2001

STATUS

approved

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Last modified June 29 01:38 EDT 2017. Contains 288855 sequences.