cp's OEIS Frontend

A227610 Number of ways 1/n can be expressed as the sum of three distinct unit fractions: 1/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.

%I A227610 #38 Feb 16 2025 08:33:20
%S A227610 1,6,15,22,30,45,36,62,69,84,56,142,53,124,178,118,67,191,74,274,227,
%T A227610 145,87,342,146,162,216,322,100,461,84,257,304,199,435,508,79,204,360,
%U A227610 580,115,587,98,455,618,192,129,676,217,417,369,449,119,573,543,759,367,240,166,1236,102,261,857,428,568,717,115,537,460,1018,155,1126,112,276,839
%N A227610 Number of ways 1/n can be expressed as the sum of three distinct unit fractions: 1/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z.
%C A227610 See A073101 for the 4/n conjecture due to Erdős and Straus.
%H A227610 Jud McCranie, <a href="/A227610/b227610.txt">Table of n, a(n) for n = 1..500</a>
%H A227610 Christian Elsholtz, <a href="https://doi.org/10.1090/S0002-9947-01-02782-9">Sums Of k Unit Fractions</a>, Trans. Amer. Math. Soc. 353 (2001), 3209-3227.
%H A227610 David Eppstein, <a href="http://www.ics.uci.edu/~eppstein/numth/egypt/intro.html">Algorithms for Egyptian Fractions</a>
%H A227610 David Eppstein, <a href="http://library.wolfram.com/infocenter/Articles/2926/">Ten Algorithms for Egyptian Fractions</a>, Wolfram Library Archive.
%H A227610 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html">Egyptian Fractions</a>
%H A227610 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EgyptianFraction.html">Egyptian Fraction</a>
%H A227610 <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%e A227610 a(1)=1 because 1 = 1/2 + 1/3 + 1/6;
%e A227610 a(2)=6 because 1/2 = 1/3 + 1/7 + 1/42 = 1/3 + 1/8 + 1/24 = 1/3 + 1/9 + 1/18 = 1/3 + 1/10 + 1/15 = 1/4 + 1/5 + 1/20 = 1/4 + 1/6 + 1/12;
%e A227610 a(3)=15 because 1/3 = 1/x + 1/y + 1/z presented as {x,y,z}: {4,13,156}, {4,14,84}, {4,15,60}, {4,16,48}, {4,18,36}, {4,20,30}, {4,21,28}, {5,8,120}, {5,9,45}, {5,10,30}, {5,12,20}, {6,7,42}, {6,8,24}, {6,9,18}, {6,10,15}; etc.
%t A227610 f[n_] := Length@ Solve[1/n == 1/x + 1/y + 1/z && 0 < x < y < z, {x, y, z}, Integers]; Array[f, 70]
%Y A227610 Cf. A002966, A073546.
%Y A227610 Cf. A227611 (2/n), A075785 (3/n), A073101 (4/n), A075248 (5/n), A227612.
%Y A227610 Cf. A347566, A347569.
%K A227610 nonn
%O A227610 1,2
%A A227610 _Robert G. Wilson v_, Jul 17 2013