This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A216328 #16 Sep 28 2024 16:16:24
%S A216328 243,70225,265879,953125,1015623,1071873,1922373,6436341,6739605,
%T A216328 7263025
%N A216328 Values for b in abc-triples with a = 2.
%C A216328 The listed 10 b-values are the ones for all (2,b,2+b) triples
%C A216328 with b from the range {1, 2, ..., 10^7}. The best quality among these values appears for n=8: (2, 6436341, 6436343), b = 3^10*109, with rad(2*b*(2+b)) = 15042 =2*3*23*109 and q(2,6436341,6436343) = 1.629911684 (maple 10 digits). See Tabl. I of the (not updated) link: The ABC Conjecture Home Page.
%C A216328 See A216323 for the list of increasing b values for abc-triples if a=1. There one finds also a reference and a maple program which can be adapted to a=2 instead of a=1.
%C A216328 This sequence is infinite because it contains the infinite subsequence b(k) = 243^(84k+1), k >= 0. - _William Hu_, Aug 29 2024
%H A216328 The ABC Conjecture Home Page, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html#Ten%20abc">The top ten good abc-examples</a>.
%F A216328 (2, b=a(n), 2+a(n)) is an abc-triple (which has quality q > 1) with increasingly ordered b values. See the comment above for abc-triples.
%e A216328 n: (a=2, b, c=2+a), rad(a*b*c), q(a*b*c) (maple 10 digits)
%e A216328 1: (2, 243, 245), 210, 1.028828797
%e A216328 2: (2, 70225, 70227), 27030, 1.093563284
%e A216328 3: (2, 255879, 255881), 252642, 1.001024059
%e A216328 4: (2, 953125, 953127), 525210, 1.045245231
%e A216328 5: (2, 1015623, 1015625), 128310, 1.175886268
%e A216328 6: (2, 1071873, 1071875), 926310, 1.010623492
%e A216328 7: (2, 1922373, 1922375), 799890, 1.064510569
%e A216328 8: (2, 6436341, 6436343), 15042, 1.629911684
%e A216328 9: (2, 6739605, 6739607), 3621030, 1.041135746
%e A216328 10: (2, 7263025, 7263027), 94710, 1.378732296
%e A216328 ...
%e A216328 From _Wolfdieter Lang_, Oct 02 2012: (Start)
%e A216328 The prime number decomposition of the ten b-values is
%e A216328 3^5, 5^2*53^2, 3^9*13, 5^6*61, 3^2*7^4*47, 3^5*11*401, 3^8*293, 3^10*109, 3^6*5*43^2, 5^2*7^4*11^2.
%e A216328 The ten c = b+2 numbers have the prime number decomposition
%e A216328 5*7^2, 3^5*17^2, 41*79^2, 3^4*7*41^2, 5^7*13, 5^5*7^3, 5^3*7*13^3, 23^5, 7^5*401, 3^11*41. (End)
%p A216328 See the program given in A216323, adapted to a=2.
%Y A216328 Cf. A216323, A216370.
%K A216328 nonn,more
%O A216328 1,1
%A A216328 _Wolfdieter Lang_, Sep 28 2012