cp's OEIS Frontend

A198443 Conjectured 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).

%I A198443 #19 Sep 25 2023 09:31:50
%S A198443 3,4,11,26,37,368,1828,2180,7825,8177,8217,71393,72481,75154,118409,
%T A198443 175485,203697,206370,1049148,1058224,1843945,1846618,8186369,8197633,
%U A198443 9600802,96020524,169503449,294638801,305158594,305192969,657099024
%N A198443 Conjectured 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 A198443 Distance d is equal to 0 when x = k^2 and y = k^5.
%C A198443 Only the values of x < 10^8 have been searched/
%C A198443 For x values see A198444.
%C A198443 For y values see A198445.
%C A198443 Conjecture: For any positive number x >= A198444(n), the distance d between the square of an integer y and the fifth power of a positive integer x (such that x <> k^2 and y <> k^5) can't be less than A198443(n).
%F A198443 a(n) = (A198445(n))^2 - (A198444(n))^5.
%t A198443 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}]; dd
%Y A198443 Cf. A179406, A179407, A179408, A198445.
%K A198443 nonn,hard
%O A198443 1,1
%A A198443 _Artur Jasinski_, Oct 25 2011