This site is supported by donations to The OEIS Foundation.

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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. 26-34. 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.