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Search: id:A018892
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| A018892 |
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Number of ways to write 1/n as a sum of exactly 2 unit fractions. |
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18
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| 1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 14, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 14, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 23, 2, 5, 8, 7, 5, 14, 2, 8, 5, 14, 2, 18, 2, 5, 8, 8, 5, 14, 2, 14, 5, 5, 2, 23, 5, 5, 5, 11, 2, 23, 5, 8, 5, 5, 5, 17, 2, 8, 8
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) =(tau(n^2)+1)/2. Number of elements in the set {(x,y): x|n, y|n, x<=y, GCD(x,y)=1}. Number of divisors of n^2 less than or equal to n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 03 2002
Equivalently number of pairs (x,y) such that LCM(x,y)=n - Benoit Cloitre (benoit7848c(AT)orange.fr), May 16 2002
Number of right triangles with an integer hypotenuse and height n. - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jul 10 2002
Number of solutions to x^3==n^2 (mod x) 1<=x<=n - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 19 2002
Except for the initial term, each entry is at least equal to 2 because of the identities 1/n=1/2n + 1/2n=1/(n+1) + 1/n*(n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), May 04 2004
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REFERENCES
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Kevin Brown, Posting to netnews group sci.math, Aug 17 1996.
L. E. Dickson, History of The Theory of Numbers, Vol. 2 p. 690, Chelsea NY 1923.
Problem 1051(a), American Mathematical Monthly, Vol. 105, No. 4, 1998 p. 372.
A. M. & I. M. Yaglom, Challenging Mathematical Problems With Elementary Solutions, Vol. 1 pp. 8;60 Prob. 19 Dover NY
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892
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FORMULA
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If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2 a1 + 1)(2 a2 + 1) ... (2 at + 1) + 1)/2.
a(n)=A063647(n)+1=A046079(2n)+1. - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 01 2003
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EXAMPLE
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Examples:
n=1: 1/1.
n=2: 1/2 = 1/4 + 1/4 = 1/3 + 1/6.
n=3: 1/3 = 1/6 + 1/6 = 1/4 + 1/12.
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MATHEMATICA
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f[j_, n_] := (DivisorSigma[0, n^j] + j - 1)/j; Table[ f[n, n], {n, 96}] (* or *)
f[j_, n_] := (Times @@ (j(Last /@ FactorInteger[n]) + 1) + j - 1)/j; Table[f[2, n], {n, 96}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 03 2005)
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PROGRAM
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(PARI) A018892(n)=(numdiv(n^2)+1)/2 - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 30 2007
(PARI) A018892s(n)=local(t=divisors(n^2)); vector((#t+1)/2, i, [n+t[i], n+n^2/t[i]]) /* show solutions */ - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 30 2007
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CROSSREFS
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Records: A126097, A126098. Cf. A048691, A063647.
Sequence in context: A102247 A054249 A033099 this_sequence A100565 A010846 A073023
Adjacent sequences: A018889 A018890 A018891 this_sequence A018893 A018894 A018895
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net), Sep 15, 1996
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