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A192789 Number of distinct solutions of 4/p = 1/a + 1/b + 1/c in positive integers satisfying 1<=a<=b<=c where p is the n-th prime.

%I A192789 #47 Nov 22 2017 15:59:07
%S A192789 1,3,2,7,9,4,4,11,21,7,19,9,7,14,34,13,27,11,17,40,7,37,27,10,8,16,27,
%T A192789 25,15,13,33,32,17,36,18,31,24,24,65,26,47,17,67,6,23,42,30,58,37,20,
%U A192789 19,106,8,51,19,71,28,48,31,17,33,34,40,79,16,34,38,21,39,32,19,110,52,33,39,86,30,29,23,15,81,16,93,19
%N A192789 Number of distinct solutions of 4/p = 1/a + 1/b + 1/c in positive integers satisfying 1<=a<=b<=c where p is the n-th prime.
%C A192789 The Erdos-Straus conjecture is equivalent to the conjecture that a(n) > 0 for all n.
%H A192789 Jean-François Alcover, <a href="/A192789/b192789.txt">Table of n, a(n) for n = 1..1000</a>
%H A192789 Christian Elsholtz, Terence Tao, <a href="https://arxiv.org/abs/1107.1010">Counting the number of solutions to the Erdos-Straus equation on unit fractions</a>, Aug 2, 2015, arXiv:1107.1010 [math.NT], 2011-2015.
%H A192789 Elsholtz, C., Tao, T. <a href="https://doi.org/10.1017/S1446788712000468">Counting the number of solutions to the Erdos-Straus equation on unit fractions</a>. Journal of the Australian Mathematical Society, 94(1), 50-105, 2013.
%H A192789 doi:10.1017/S1446788712000468
%F A192789 a(n) = A192787(prime(n)). - _Michel Marcus_, Aug 20 2014
%e A192789 a(1) = 1 because 4/prime(1) = 1/1 + 1/2 + 1/2.
%p A192789 a:= n-> A192787(ithprime(n)):
%p A192789 seq(a(n), n=1..70);
%t A192789 a[n_] := a[n] = Module[{a, b, c, r}, r = Reduce[1 <= a <= b <= c && 4/Prime[n] == 1/a + 1/b + 1/c, {a, b, c}, Integers]; Which[Head[r] === Or, Length[r], Head[r] === And, 1, r === False, 0, True, Print["error: ", r]]];
%t A192789 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 84}] (* _Jean-François Alcover_, Nov 22 2017 *)
%o A192789 (PARI) a(n)=my(t=4/prime(n), t1, s, c); for(a=1\t+1, 3\t, t1=t-1/a; for(b=1\t1+1, 2\t1, c=1/(t1-1/b); if(denominator(c)==1&&c>=b, s++))); s
%Y A192789 A292624 counts the solutions with multiplicity.
%Y A192789 Cf. A192787, A292581, A292581.
%K A192789 nonn
%O A192789 1,2
%A A192789 _Charles R Greathouse IV_, Jul 10 2011