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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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19
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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; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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.rs), 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)gmail.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
| K. S. 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
| 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
| 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
| Examples:
n=1: 1/1 = 1/2 + 1/2.
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
| 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)
a[n_] := (DivisorSigma[0, n^2] + 1)/2; Table[a[n], {n, 1, 99}](* From Jean-François Alcover, Dec 19 2011, after Vladeta Jovovic *)
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PROG
| (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
(Haskell)
a018892 n = length [d | d <- [1..n], n^2 `mod` d == 0]
-- Reinhard Zumkeller, Jan 08 2012
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CROSSREFS
| Records: A126097, A126098. Cf. A048691, A063647.
Sequence in context: A160273 A141822 A033099 * A100565 A010846 A073023
Adjacent sequences: A018889 A018890 A018891 * A018893 A018894 A018895
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net), Sep 15, 1996
First example corrected by Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 02 2009
Incorrect Mathematica program deleted by N. J. A. Sloane, Jul 08 2009
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