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A216323
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Values for b in abc-triples with a=1.
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3
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8, 48, 63, 80, 224, 242, 288, 512, 624, 675, 728, 960, 1024, 1215, 2303, 2400, 3024, 3887, 3968, 4095, 4374, 5831, 6399, 6560, 6655, 6859, 8575, 9375, 9408, 9800, 10647, 12167, 14336, 15624, 16128, 17576, 21951, 24299, 25920, 28125, 29375, 29791
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OFFSET
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1,1
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COMMENTS
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For abc-triples see de Smit's link.
(a, b, c=a+b) with positive integers a and b, a <= b, gcd(a,b) = 1 is called an abc-triple if r(a,b,c) < c where r(a,b,c) = rad(a*b*c) with rad = A007947 (radical or squarefree kernel). The quality q of an abc-triple is the real positive number q(a,b,c) = log(c)/log(r(a,b,c)), hence q > 1. See also a comment on A216370.
Here one considers a = 1, c = 1+b for b >= 1.
The radical r(1,a(n),a(n)+1) for these abc-triples is 2*A216324.
The highest quality q of the 258 abc-triples (1, a(n), a(n)+1) with b in the range 1..10^7 appears for the triple (1, 4374, 4375) with b = a(21) and q = 1.567887264 (maple 10 digits).
This sequence is infinite because it contains the infinite subsequence b(k) = 9^k - 1, k>=1.
Alvarez-Salazar et al. prove that k is a term iff k/rad(k) > rad(k+1). - Michel Marcus, Jan 05 2023
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LINKS
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FORMULA
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(1, b=a(n), a(n)+1) is an abc-triple (which has quality q > 1) with increasingly ordered b values. See the comment above for abc-triples.
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MAPLE
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read "abc1bN.txt": abc1bN(30000); (with the above given maple text file).
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MATHEMATICA
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rad[n_] := Times @@ Transpose[FactorInteger[n]][[1]]; a = 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, b]], {b, a, mx - a}], {n, 5}]; t (* T. D. Noe, Sep 24 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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