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Search: id:A063647
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| A063647 |
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Number of ways to write 1/n as a difference of exactly 2 unit fractions. |
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18
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| 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; listen)
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OFFSET
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1,4
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COMMENT
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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 (vladeta(AT)eunet.rs), 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 (amarnath_murthy(AT)yahoo.com), Dec 24 2001
Number of elements in the set {(x,y): x|n, y|n, x<y, GCD(x,y)=1}. - Vladeta Jovovic (vladeta(AT)eunet.rs), May 03 2002
Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (ghodges14(AT)comcast.net), Aug 25 2004
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REFERENCES
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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.
Crux Mathematicorum, CMS Vol. 23 No. 7 Nov. 1997 pp. 443-4 Soln. Prob. 2175.
Problem 1051(b), American Mathematical Monthly, Vol. 105, No. 4, 1998 p. 372.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
Canadian Math. Soc., "Crux Mathematicorum", Vol. 23, No. 7, Nov.,1997, pp 443-4 Soln. to Prob. 2175
M. L. Perez et al., eds., Smarandache Notions Journal
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FORMULA
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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*ln(n)*ln(ln(n))+o(ln(n)) with C=0.7...
Bisection of A046079. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 09 2004
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EXAMPLE
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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).
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MATHEMATICA
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Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
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PROGRAM
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(PARI) for(n=1, 100, print1(sum(i=1, n^2, if((n*i)%(i+n), 0, 1)), ", "))
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CROSSREFS
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Cf. A018892, A063427, A063428. First twenty-nine terms identical to those of A062799 (offset).
Cf. A063717, A063718, A048691.
Sequence in context: A067614 A113901 A062799 this_sequence A077808 A021471 A088372
Adjacent sequences: A063644 A063645 A063646 this_sequence A063648 A063649 A063650
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Jul 23 2001
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