A276306 Number of pairs of integers (k, m) with k < m < n such that (k, m, n) is an abc-triple.

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 3

Author

Felix Fröhlich, Aug 29 2016

Keywords

• nonn

Comments

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.

a(n) > 0 for n in A120498.

a(n) gives the number of times n appears in A130510.

a(n) gives the number of i such that A225426(A008585(i)) = n.

Examples

			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.
		

Links

• Wikipedia, abc conjecture.

Crossrefs

Cf. A120498, A130510, A225426.

Programs

Mathematica

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 *)

PARI

rad(x, y, z) = my(f=factor(x*y*z)[, 1]~); prod(i=1, #f, f[i])
is_abc_hit(x, y, z) = z==x+y && gcd(x, y)==1 && rad(x, y, z) < z
a(n) = my(i=0); for(m=1, n-1, for(k=1, m-1, if(is_abc_hit(k, m, n), i++))); i