A216370 Number of ABC triples with quality q > 1 and c < 10^n.

1, 6, 31, 120, 418, 1268, 3499, 8987, 22316, 51677, 116978, 252856, 528275, 1075319, 2131671, 4119410, 7801334, 14482059

Offset: 1

Author

Jonathan Vos Post, Sep 05 2012

Keywords

• nonn
• more

Examples

			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}.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 2004, ISBN 0-387-20860-7.
• Carl Pomerance, Computational Number Theory, The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361-362.

Links

• Jordan Ellenberg, Mochizuki on ABC, Sep 03 2012.
• Dorian Goldfeld, Beyond the last theorem, Math Horizons, 1996 (September), pp. 26-34.
• Reken Mee met ABC, Synthese resultaten, (Dutch), 2011.
• Wikipedia, abc conjecture

Crossrefs

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

Programs

Mathematica

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 *)