A375019 -id:A375019 - cp's OEIS frontend

A375622 Numbers k such that k gives the maximum quality for abc triples of the form (1, k^n-1, k^n) where n is the sequence index.

4375, 49, 361, 7, 9, 19, 129, 7, 28, 3, 243, 19, 625, 26, 9, 3

Offset: 1

Frank M Jackson, Aug 21 2024

An abc triple is defined as (a, b, c) with a + b = c, gcd(a, b) = 1 and radical of a*b*c, rad(a*b*c) < c. The quality of an abc triple is q = log(c)/log(rad(a*b*c)). For each n, a sample of the first 100 abc triples of the form (1, k^n-1, k^n) is compared to find the value of k that gives the abc triple maximum quality. The sample size s = 100 of abc triples appears adequate to identify the maximum quality because the quality term tends rapidly towards the lim sup(q) = 1 as s -> oo.

			a(2) = 49 because from the sample of 100 abc triples of the form (1, k^2-1, k^2) (see A375019) where k takes values 3, 7,..., 49,..., 3362, 3375, when k = 49 = A375019(12), we get maximum quality q = 1.45567... with triple (1, 2400, 2401).

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.

Cf. A007947, A216323, A375019.

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; Table[(lst={}; k=2; While[Length@lst<100, If[Rad[(k^n-1)*k]

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A375622 Numbers k such that k gives the maximum quality for abc triples of the form (1, k^n-1, k^n) where n is the sequence index.

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