A216370 - OEIS

%I #30 Jan 05 2023 10:17:23

%S 1,6,31,120,418,1268,3499,8987,22316,51677,116978,252856,528275,

%T 1075319,2131671,4119410,7801334,14482059

%N Number of ABC triples with quality q > 1 and c < 10^n.

%D Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 2004, ISBN 0-387-20860-7.

%D Carl Pomerance, Computational Number Theory, The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361-362.

%H Jordan Ellenberg, <a href="http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/">Mochizuki on ABC</a>, Sep 03 2012.

%H Dorian Goldfeld, <a href="https://www.jstor.org/stable/25678079">Beyond the last theorem</a>, Math Horizons, 1996 (September), pp. 26-34.

%H Reken Mee met ABC, <a href="https://web.archive.org/web/20160311110515/http://www.rekenmeemetabc.nl/?item=h_stats">Synthese resultaten</a>, (Dutch), 2011.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>

%e a(2) = 6 because there are 6 (a,b,c) triples with c < 10^2 and q > 1. Those triples are {1,8,9}, {1,48,49}, {1,63,64}, {1,80,81}, {5,27,32}, and {32,49,81}.

%t rad[n_] := Times @@ Transpose[FactorInteger[n]][[1]]; Table[t = {}; mx = 10^n; Do[c = a + b; If[c < mx && GCD[a, b] == 1 && Log[c] > Log[rad[a*b*c]], AppendTo[t, {a, b, c}]], {a, mx/2}, {b, a, mx - a}]; Length[t], {n, 3}] (* _T. D. Noe_, Sep 06 2012 *)

%Y Cf. A007947, A120498, A130510, A130511, A130512.

%K nonn,more

%O 1,2

%A _Jonathan Vos Post_, Sep 05 2012