login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,4

COMMENTS

In 1948 Erdős 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.

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

P. Erdős, Az 1/z_1 + 1/z_2 + ... + 1/z_n = a/b egyenlet egész számú megoldásairól, (On a Diophantine equation), Mat. Lapok, 1:192-210, 1050. Math. Rev. 13:208b.

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.

MAPLE

A:= proc(n)

   local x, t, p, q, ds, zs, ys, js, tot, j;

tot:= 0;

for x from 1+floor(n/4) to ceil(3*n/4)-1 do

    t:= 4/n - 1/x;

    p:= numer(t);

    q:= denom(t);

    ds:= convert(select(d -> (d < q) and d + q mod p = 0,

          numtheory:-divisors(q^2)), list);

    ys:= map(d -> (d+q)/p, ds);

    zs:= map(d -> (q^2/d+q)/p, ds);

    js:= select(j -> ys[j] > x, [$1..nops(ds)]);

    tot:= tot + nops(js);

od;

tot;

end proc:

seq(A(n), n=2..100); # Robert Israel, Aug 22 2014

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,changed

AUTHOR

Robert G. Wilson v, Aug 18 2002

EXTENSIONS

Edited by T. D. Noe, Sep 10 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 22 06:43 EST 2014. Contains 249804 sequences.