A216323 Values for b in abc-triples with a=1.

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

Offset: 1

Author

Wolfdieter Lang, Sep 24 2012

Keywords

• nonn

Comments

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

Links

• Frank M Jackson, Table of n, a(n) for n = 1..1500
• Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku, and Summer Soller, On abc triples of the form (1,c-1,c), arXiv:2301.01376 [math.NT], 2023.
• Bart de Smit, Triples of small size [references the ABC@Home project which is inactive since 2015]
• Wolfdieter Lang, Maple program abc1bN.txt for A216323 for b in the range 1..N .

Crossrefs

Cf. A007947, A216324, A216370.

Programs

Maple

read "abc1bN.txt":  abc1bN(30000); (with the above given maple text file).

Mathematica

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 *)
Rad[n_] := Module[{lst = FactorInteger[n]}, Times @@ (First /@ lst)]; lst={};
n = 1; While[Length@lst <= 10^2, If[n/Rad[n]>Rad[n+1], AppendTo[lst, n]]; n++]; lst (* Frank M Jackson, Sep 04 2024 *)

Formula

(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.