%I #83 Jul 03 2022 09:17:36

%S 0,1,3,3,2,8,7,10,6,12,9,21,4,17,39,28,4,26,11,36,29,25,21,57,10,20,

%T 29,42,7,81,19,70,31,25,65,79,9,32,73,96,7,86,14,62,93,42,34,160,18,

%U 53,52,59,13,89,98,136,41,33,27,196,11,37,155,128,49,103,17,73,55,185,40,211,7,32,129,80,97,160,37,292

%N Number of distinct solutions of 4/n = 1/a + 1/b + 1/c in positive integers satisfying 1 <= a <= b <= c.

%C The Erdős-Straus conjecture is that a(n) > 0 for n > 1. Swett verified the conjecture for n < 10^14.

%C Vaughan shows that the number of n < x with a(n) = 0 is at most x exp(-c * (log x)^(2/3)) for some c > 0.

%C See A073101 for the 4/n conjecture due to Erdős and Straus.

%H Charles R Greathouse IV, <a href="/A192787/b192787.txt">Table of n, a(n) for n = 1..10000</a>

%H Christian Elsholtz, Terence Tao, <a href="http://arxiv.org/abs/1107.1010">Counting the number of solutions to the Erdős-Straus equation in unit fractions</a>, arXiv:1107.1010 [math.NT], 2010-2015.

%H Maria Monks, Ameya Velingker, <a href="https://web.archive.org/web/20150913002803/http://www.cs.cmu.edu/afs/cs.cmu.edu/Web/People/avelingk/papers/erdos_straus.pdf">On the Erdős-Straus conjecture: Properties of solutions to its underlying diophantine equation</a>. See <a href="https://www.semanticscholar.org/paper/On-the-Erd%C3%B6s-Straus-conjecture-%3A-Properties-of-to-Monks-Velingker/b65e60f1528dfc9751ee4d7b3240dd6dd8e3fbc2">also</a>.

%H Alan Swett, <a href="https://web.archive.org/web/20110716110731/http://math.uindy.edu/swett/esc.htm">The Erdos-Strauss Conjecture</a>, 1999.

%H R. C. Vaughan, <a href="http://dx.doi.org/10.1112/S0025579300002886">On a problem of Erdős, Straus and Schinzel</a>, Mathematika 17 (1970), pp. 193-198.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture">Erdős-Straus conjecture</a>.

%H Konstantine Zelator, <a href="http://arxiv.org/abs/0912.2458">An ancient Egyptian problem: the diophantine equation 4/n=1/x+1/y+1/z, n>or=2</a>, arXiv:0912.2458 [math.GM], 2009.

%e a(1) = 0, since 4/1 = 4 cannot be expressed as the sum of three reciprocals.

%e a(2) = 1 because 4/2 = 1/1 + 1/2 + 1/2, and there are no other solutions.

%e a(3) = 3 since 4/3 = 1 + 1/4 + 1/12 = 1 + 1/6 + 1/6 = 1/2 + 1/2 + 1/3.

%e a(4) = 3 = A002966(3).

%p A192787 := proc(n) local t,a,b,t1,count; t:= 4/n; count:= 0; for a from floor(1/t)+1 to floor(3/t) do t1:= t - 1/a; for b from max(a,floor(1/t1)+1) to floor(2/t1) do if type( 1/(t1 - 1/b),integer) then count:= count+1; end if end do end do; count; end proc; # _Robert Israel_, Feb 19 2013

%t f[n_] := Length@ Solve[ 4/n == 1/x + 1/y + 1/z && 1 <= x <= y <= z, {x, y, z}, Integers]; Array[f, 70] (* _Allan C. Wechsler_ and _Robert G. Wilson v_, Jul 19 2013 *)

%o (PARI) a(n, show=0)=my(t=4/n, t1, s, c); for(a=1\t+1, 3\t, t1=t-1/a; for(b=max(1\t1+1, a), 2\t1, c=1/(t1-1/b); if(denominator(c)==1&&c>=b, s++; show&&print("4/",n," = 1/",a," + 1/",b," + 1/",c)))); s \\ variant with print(...) added by _Robert Munafo_, Feb 19 2013, both combined through option "show" by _M. F. Hasler_, Jul 02 2022

%Y A292581 counts the solutions with multiplicity. A073101 counts solutions with a, b, and c distinct.

%Y Cf. A292581, A292624, A192789, A075245, A075246, A075247, A075248.

%Y Cf. A004194, A226641, A226642, A226644, A226645, A226646.

%Y Cf. A337432 (solutions with minimal c).

%K nonn

%O 1,3

%A _Charles R Greathouse IV_, Jul 10 2011

%E Corrected at the suggestion of _Allan C. Wechsler_ by _Charles R Greathouse IV_, Feb 19 2013

%E Examples and cross-references added by _M. F. Hasler_, Feb 19 2013