

A216370


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


2



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

1,2


COMMENTS

The abc conjecture (or OesterlĂ©Masser conjecture) is a conjecture in number theory, first proposed by Joseph OesterlĂ© and David Masser in 1985, stated in terms of three positive integers, a, b and c (whence the name), which have no common factor and satisfy a + b = c. Quantity q = log(c)/log(rad(a*b*c)) where rad(k) = A007947(k), is the product of the distinct prime factors of k. This is the q > 1 column of the wikipedia table taken from RekenMeeMetABC.nl (2011). Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis." Jordan Ellenberg at Quomodocumque reports on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki.


REFERENCES

Dorian Goldfeld, Beyond the last theorem, Math Horizons, 1996 (September), pp. 2634.
Richard K. Guy, Unsolved Problems in Number Theory, SpringerVerlag, 2004, ISBN 0387208607.
Carl Pomerance, Computational Number Theory, The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361362.


LINKS

Table of n, a(n) for n=1..18.
Jordan Ellenberg, Mochizuki on ABC
Reken Mee met ABC, Synthese resultaten, (Dutch), 2011
Wikipedia, abc conjecture


EXAMPLE

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}.


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 *)


CROSSREFS

Cf. A007947, A120498, A130510, A130511, A130512.
Sequence in context: A143568 A166786 A003728 * A225425 A267890 A244199
Adjacent sequences: A216367 A216368 A216369 * A216371 A216372 A216373


KEYWORD

nonn


AUTHOR

Jonathan Vos Post, Sep 05 2012


STATUS

approved



