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%I #11 Mar 24 2017 00:47:59
%S 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0
%N Number of pairs of integers (k, m) with k < m < n such that (k, m, n) is an abc-triple.
%C An abc-triple is a set of three integers (a, b, c) such that a+b = c, gcd(a, b) = 1 and rad(a, b, c) < c, where rad() gives the product of the distinct prime factors of its arguments.
%C a(n) > 0 for n in A120498.
%C a(n) gives the number of times n appears in A130510.
%C a(n) gives the number of i such that A225426(A008585(i)) = n.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>.
%e For n = 81: there are 2 abc-triples for c = 81 with a < b < c, namely (32, 49, 81) and (1, 80, 81), so a(81) = 2.
%t rad[a_, b_, c_] := Times @@ FactorInteger[a b c][[All, 1]]; abcTripleQ[a_, b_, c_] := a + b == c && GCD[a, b] == 1 && rad[a, b, c] < c; a[n_] := (For[i = 0; m = 1, m <= n-1, m++, For[k = 1, k <= m-1, k++, If[ abcTripleQ[k, m, n], i++]]]; i); Table[a[n], {n, 3, 89}] (* _Jean-François Alcover_, Sep 04 2016, partly adapted from PARI *)
%o (PARI) rad(x, y, z) = my(f=factor(x*y*z)[, 1]~); prod(i=1, #f, f[i])
%o is_abc_hit(x, y, z) = z==x+y && gcd(x, y)==1 && rad(x, y, z) < z
%o a(n) = my(i=0); for(m=1, n-1, for(k=1, m-1, if(is_abc_hit(k, m, n), i++))); i
%Y Cf. A120498, A130510, A225426.
%K nonn
%O 3,79
%A _Felix Fröhlich_, Aug 29 2016
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