A375622 - OEIS

%I #11 Sep 05 2024 03:27:53

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

%N 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.

%C 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.

%H Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku, and Summer Soller, <a href="https://arxiv.org/abs/2301.01376">On abc triples of the form (1,c-1,c)</a>, arXiv:2301.01376 [math.NT], 2023.

%e 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).

%t 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}]

%Y Cf. A007947, A216323, A375019.

%K nonn,more

%O 1,1

%A _Frank M Jackson_, Aug 21 2024