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A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.

%I A376925 #57 Dec 23 2024 15:11:45
%S A376925 2,9,128,4375,18225,1771561,27295744,1375325568,6313843404,
%T A376925 289478389760
%N A376925 a(n) is the largest number that can be written as x + y with x and y coprime and such that each of x, y, and x + y are prime(n)-smooth.
%C A376925 The terms given here can be verified by checking that the number of solutions up to a(n) equals A362593(n) and a(n) is a solution x+y.
%C A376925 a(2) = 9 corresponds to Catalan's conjecture (Mihăilescu's theorem).
%C A376925 a(4) = 4375 corresponds to the final term of A303332.
%H A376925 Wikipedia, <a href="https://en.wikipedia.org/wiki/Abc_conjecture">Abc conjecture</a>.
%H A376925 Wikipedia, <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture">Catalan's conjecture</a>.
%e A376925 a(3) = 128 because prime(3) = 5, and 125 + 3 = 128 with 125 and 3 coprime, and 125, 3 and 128 are all 5-smooth numbers, and no number larger than 128 has these properties.
%e A376925 Table x + y = a(n) is shown below (q gives abc triple quality):
%e A376925   n=1: 1 + 1 = 2 (q=1),
%e A376925   n=2: 8 + 1 = 9 (q=1.226)
%e A376925   n=3: 125 + 3 = 128 (q=1.426)
%e A376925   n=4: 4374 + 1 = 4375 (q=1.567)
%e A376925   n=5: 14641 + 3584 = 18225 (q=1.267)
%e A376925   n=6: 1771470 + 91 = 1771561 (q=1.395)
%e A376925   n=7: 27217619 + 78125 = 27295744 (q=1.421)
%e A376925   n=8: 1371299293 + 4026275 = 1375325568 (q=1.31)
%e A376925   n=9: 4867359029 + 1446484375 = 6313843404 (q=1.17)
%e A376925   n=10: 289478257991 + 131769  = 289478389760 (q=1.16)
%Y A376925 Cf. A303332, A362593.
%K A376925 nonn,more
%O A376925 1,1
%A A376925 _Zhicheng Wei_, Oct 10 2024
%E A376925 a(7)-a(8) from _Andrew Howroyd_, Oct 12 2024
%E A376925 a(9)-a(10) from _David A. Corneth_, Nov 24 2024