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A130510 ABC conjecture: values of c in the list of "abc-hits".

%I A130510 #17 Oct 19 2023 22:31:48
%S A130510 9,32,49,64,81,81,125,128,225,243,245,250,256,256,289,343,375,512,512,
%T A130510 513,539,625,625,625,676,729,729,729,729,961,968,1025,1029,1216,1331,
%U A130510 1331,1331,1369,1587,1681,2048,2048,2048,2057,2187,2187,2187,2197,2197
%N A130510 ABC conjecture: values of c in the list of "abc-hits".
%C A130510 Let rad(x) be the function that computes the squarefree kernel of x (see A007947). A triple {a,b,c} of positive integers with a+b=c, gcd(a,b)=1 and c > rad(a*b*c) is called an abc-hit. The corresponding values of a and rad(a*b*c) are in the sequences A130511 and A130512.
%D A130510 See A120498
%H A130510 T. D. Noe, <a href="/A130510/b130510.txt">Table of n, a(n) for n=1..1269</a> (for c up to 10^6)
%H A130510 Sander R. Dahmen, <a href="http://dx.doi.org/10.1016/j.jnt.2007.06.009">Lower bounds for numbers of ABC-hits</a>, J. Numb. Theory, Volume 128, Issue 6, June 2008, pp. 1864-1873.
%H A130510 Noam D. Elkies, <a href="http://www.math.harvard.edu/hcmr/issues/1.pdf">The ABC's of Number Theory</a>, The Harvard College Mathematics Review, Vol. 1, No. 1, Spring 2007, pp. 57-76.
%H A130510 Brian Hayes, <a href="http://bit-player.org/2007/easy-as-abc">Easy as abc</a>
%H A130510 Wikipedia, <a href="http://en.wikipedia.org/wiki/Abc_conjecture">abc conjecture</a>
%e A130510 81 appears twice because 1+80=81 and 32+49=81 are two abc-hits.
%t A130510 rad[n_] := If[n==1, 1, Times@@(Transpose[FactorInteger[n]][[1]])]; nn=1000; Do[If[ !PrimeQ[c], Do[b=c-a; If[GCD[a,b]==1 && rad[a*b*c]<c, Print[{a,b,c,rad[a*b*c]}]], {a,c/2}]], {c,2,nn}]
%o A130510 (Python)
%o A130510 from itertools import count, islice
%o A130510 from math import prod, gcd
%o A130510 from sympy import primefactors
%o A130510 def A130510_gen(startvalue=1): # generator of terms >= startvalue
%o A130510     for c in count(max(startvalue,1)):
%o A130510         pc = set(primefactors(c))
%o A130510         for a in range(1,(c>>1)+1):
%o A130510             b = c-a
%o A130510             if gcd(a,b)==1 and c>prod(set(primefactors(a))|set(primefactors(b))|pc):
%o A130510                 yield c
%o A130510 A130510_list = list(islice(A130510_gen(),30)) # _Chai Wah Wu_, Oct 19 2023
%Y A130510 Cf. A120498 (unique values of c).
%Y A130510 Cf. A130511, A130512 (a, and rad(a*b*c)).
%Y A130510 Cf. A225425 (number of solutions with c < 10^n).
%Y A130510 Cf. A225426 (triples of numbers a,b,c).
%K A130510 nonn
%O A130510 1,1
%A A130510 _T. D. Noe_, Jun 01 2007