A179145 - OEIS

%I #24 Aug 24 2022 20:36:20

%S 27,125,216,1728,2197,3375,4913,6859,8000,13824,19683,24389,27000,

%T 29791,59319,68921,74088,79507,91125,103823,110592,132651,140608,

%U 148877,157464,166375,195112,205379,216000,226981,238328,287496,300763,314432

%N Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 1 integral solution.

%H Jianing Song, <a href="/A179145/b179145.txt">Table of n, a(n) for n = 1..115</a> (using the b-file of A356720, which is based on the data from A103254)

%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

%F a(n) = A356709(n)^3. - _Jianing Song_, Aug 24 2022

%t (* Assuming every term is a cube *) xmax = 2000; r[n_] := Reap[ Do[ rpos = Reduce[y^2 == x^3 + n, y, Integers]; If[rpos =!= False, Sow[rpos]]; rneg = Reduce[y^2 == (-x)^3 + n, y, Integers]; If[rneg =!= False, Sow[rneg]], {x, 1, xmax}]]; ok[n_] := Which[ rn = r[n]; rn[[2]] === {}, False, Length[rn[[2]]] > 1, False, ! FreeQ[rn[[2, 1]], Or], False, True, True]; ok[n_ /; !IntegerQ[n^(1/3)]] = False; ok[1]=False; A179145 = Reap[ Do[ If[ok[n], Print[n]; Sow[n]], {n, 1, 320000}]][[2, 1]] (* _Jean-François Alcover_, Apr 12 2012 *)

%Y Cf. A054504, A081119, A179145-A179162, A356709.

%Y Complement of A356703 among the positive cubes.

%Y Cf. also A179163, A179419.

%K nonn

%O 1,1

%A _Artur Jasinski_, Jun 30 2010

%E Edited and extended by _Ray Chandler_, Jul 11 2010