

A073101


Number of solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.


23



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
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OFFSET

2,4


COMMENTS

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.


REFERENCES

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, 1:192210, 1050. Math. Rev. 13:208b.


LINKS

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
Ron Knott Egyptian Fractions
Eric Weisstein's World of Mathematics, Egyptian Fraction


EXAMPLE

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.


MATHEMATICA

(* 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
f[n_] := Length@ Solve[4/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 72, 2] (* Robert G. Wilson v, Jul 17 2013 *)


PROG

(Haskell)
import Data.Ratio ((%), numerator, denominator)
a073101 n = length [(x, y) 
x < [n `div` 4 + 1 .. 3 * n `div` 4], let y' = recip $ 4%n  1%x,
y < [floor y' + 1 .. floor (2*y') + 1], let z' = recip $ 4%n  1%x  1%y,
denominator z' == 1 && numerator z' > y && y > x]
 Reinhard Zumkeller, Jan 03 2011


CROSSREFS

Cf. A075245, A075246, A075247, A075248.
Sequence in context: A082084 A094236 A205444 * A235526 A130851 A130856
Adjacent sequences: A073098 A073099 A073100 * A073102 A073103 A073104


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Aug 18 2002


EXTENSIONS

Edited by T. D. Noe, Sep 10 2002


STATUS

approved



