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Search: id:A073101
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| A073101 |
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Number of solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z. |
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9
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| 0, 1, 1, 2, 5, 5, 6, 4, 9, 7, 15, 4, 14, 33, 22, 4, 21, 9, 30, 25, 22, 19, 45, 10, 17, 25, 36, 7, 72, 17, 62, 27, 22, 59, 69, 9, 29, 67, 84, 7, 77, 12, 56, 87, 39, 32, 142, 16, 48, 46, 53, 13, 82, 92, 124, 37, 30, 25, 178, 11, 34, 147, 118, 49, 94, 15, 67, 51, 176, 38, 191, 7
(list; graph; listen)
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OFFSET
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2,4
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COMMENT
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In 1948 Erdos and Straus conjectured that for any positive integer n >= 2 the equation 4/n = 1/x + 1/y +1/z has a solution with positive integers x, y and z (without the additional requirement 0 < x < y < z). All of the solutions can be printed by removing the comment symbols from the Mathematica program. For the solution (x,y,z) having the largest z value, see (A075245, A075246, A075247). See A075248 for Sierpinski's conjecture for 5/n.
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REFERENCES
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P. Erdos, Az 1/z_1 + 1/z_2 + ... + 1/z_n = a/b egyenlet egesz szam u megoldasairol (On a Diophantine equation). Mat. Lapok, 1:192-210, 1050. Math. Rev. 13:208b.
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LINKS
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T. D. Noe, Table of n, a(n) for n=2..1000
Christian Elsholtz, Sums Of k Unit Fractions
David Eppstein, Algorithms for Egyptian Fractions
Eric Weisstein's World of Mathematics, Egyptian Fraction
Ron Knott Egyptian Fractions
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EXAMPLE
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a(5)=2 because there are two solutions: 4/5 = 1/2+1/4+1/20 and 4/5 = 1/2+1/5+1/10.
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MATHEMATICA
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(* download Egypt.m from D. Eppstein's site and put it into MyOwn directory underneath Mathematica\AddOns\StandardPackages *) Needs["MyOwn`Egypt`"]; Table[ Length[ EgyptianFraction[4/n, Method -> Lexicographic, MaxTerms -> 3, MinTerms -> 3, Duplicates -> Disallow, OutputFormat -> Plain]], {n, 5, 80}]
m = 4; For[lst = {}; n = 2, n <= 100, n++, cnt = 0; xr = n/m; If[IntegerQ[xr], xMin = xr + 1, xMin = Ceiling[xr]]; If[IntegerQ[3xr], xMax = 3xr - 1, xMax = Floor[3xr]]; For[x = xMin, x <= xMax, x++, yr = 1/(m/n - 1/x); If[IntegerQ[yr], yMin = yr + 1, yMin = Ceiling[yr]]; If[IntegerQ[2yr], yMax = 2yr + 1, yMax = Ceiling[2yr]]; For[y = yMin, y <= yMax, y++, zr = 1/(m/n - 1/x - 1/y); If[y > x && zr > y && IntegerQ[zr], z = zr; cnt++; (*Print[n, " ", x, " ", y, " ", z]*)]]]; AppendTo[lst, cnt]]; lst
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CROSSREFS
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Cf. A075245, A075246, A075247, A075248.
Sequence in context: A082086 A082084 A094236 this_sequence A130851 A130856 A004095
Adjacent sequences: A073098 A073099 A073100 this_sequence A073102 A073103 A073104
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 18 2002
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EXTENSIONS
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Edited by T. D. Noe (noe(AT)sspectra.com), Sep 10 2002
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