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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..16.

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.

Example

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).

Mathematica

Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; Table[(lst={}; k=2; While[Length@lst<100, If[Rad[(k^n-1)*k]<k^n, AppendTo[lst, k]]; k++]; lst1=Table[k=lst[[j]]; r=Rad[k(k^n-1)]; {n*Log[k]/Log[r], k}, {j, 1, Length@lst}]; (Flatten@Select[lst1, #[[1]]==Max@(First /@ lst1) &])[[2]]), {n, 1, 16}]

Crossrefs

Cf. A007947, A216323, A375019.

Sequence in context: A203403 A206148 A230523 * A376924 A111345 A203874

Adjacent sequences: A375619 A375620 A375621 * A375623 A375624 A375625

Keyword

nonn,more

Author

Frank M Jackson, Aug 21 2024

Status

approved