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%I #10 Sep 08 2023 22:37:37
%S 2,6,56,2537,3788,45531,90298,110302,3120599,3280601,3878907,12325663,
%T 14055482,14645977,42923597,45730778,183164286,185898039,926295393,
%U 2054642668,44803437862,44877249113,104775699199,104939539201,414619915847,17920089051165,21146208937291,52744869326263,95361328242187,9537353527343
%N Values y of record minima of the positive distance d between the square of an integer y and the fifth power of a positive integer x such that d = y^2 - x^5 (x <> k^2 and y <> k^5).
%C Distance d is equal to 0 when x = k^2 and y = k^5.
%C For d values see A198443.
%C For x values see A198444.
%C Conjecture (_Artur Jasinski_):
%C For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of x such that x <> k^2 and y <> k^5) can't be less than A198443(n).
%t max = 1000; vecd = Table[10^100, {n, 1, max}]; vecx = Table[10^100, {n, 1, max}]; vecy = Table[10^100, {n, 1, max}]; len = 1; Do[m = Floor[(n^5)^(1/2)] + 1; k = m^2 - n^5; If[k != 0, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m], {n, 1, 100000000}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; vecy
%Y Cf. A179406, A179407, A179408, A198443, A198444.
%K nonn
%O 1,1
%A _Artur Jasinski_, Oct 25 2011
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