This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179386 #41 Oct 30 2017 12:02:13
%S A179386 2,4,7,26,28,47,49,60,63,174,207,307,7670,15336,18589,22189,37071,
%T A179386 44678,63604,64432
%N A179386 Records of minima of A154333, difference of a cube minus the next smaller square.
%C A179386 "Records of minima" here means values A154333(x) such that A154333(x') > A154333(x) for all x' > x, or equivalently, the range of m(x) = min{ A154333(x') ; x' > x }. - _M. F. Hasler_, Sep 27 2013
%C A179386 For the associated x values see A179387 (and example).
%C A179386 For the associated values y=max{ y | y^2 < x^3 }, see A179388.
%C A179386 From _Artur Jasinski_, Jul 13 2010: (Start)
%C A179386 Theorem (*Artur Jasinski*)
%C A179386 For any positive number x >= A179387(n) the distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n).
%C A179386 Proof: The number of integral points of each Mordell elliptic curve of the form x^3-y^2 = k is finite and completely computable, therefore such x can't exist.
%C A179386 (End)
%C A179386 An equivalent theorem is the following (*Artur Jasinski*): For any positive number x >= 1+A179387(n) distance between cube of x and square of any y (such that x<>n^2 and y<>n^3) can't be less than A179386(n+1). - _Artur Jasinski_, Aug 11 2010
%C A179386 Also: The range of b(n) = min { A181138(m) | m>n }. - _M. F. Hasler_, Sep 26 2013
%C A179386 Indeed, if k=A154333(x) is a member if this sequence A179386, then also k=A181138(y) for the corresponding y, and since there is no larger x' such that x'^3-y'^3 <= k, there cannot be a larger y' such that k=A181138(y') (since this y' would require a corresponding x' > x). Conversely, the same reasoning holds for "records of minima" in A181138. - _M. F. Hasler_, Sep 26 and Sep 28 2013
%H A179386 J. Calvo, J. Herranz, G. Saez, <a href="https://doi.org/10.1090/S0025-5718-09-02240-6">A new algorithm to search for small nonzero |x^3 - y^2| values</a>, Math. Comp. 78 (2009), 2435-2444.
%H A179386 Noam Elkies, <a href="http://arxiv.org/abs/math/0005139">Rational points near curves and small nonzero |x^3 - y^2| via lattice reduction</a> arXiv:math/0005139 [math.NT], 2000.
%H A179386 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]
%H A179386 J. Gebel, A. Petho, G. Zimmer, <a href="https://doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Mathematica 110 (1998), 335-367. <a href="http://www.ams.org/mathscinet-getitem?mr=1602064">MR1602064</a>
%e A179386 For numbers x > 32, A154333(x) > 7.
%e A179386 For numbers x > 35, A154333(x) > 26.
%e A179386 For numbers x > 37, A154333(x) > 28.
%e A179386 For numbers x > 63, A154333(x) > 47.
%e A179386 For numbers x > 65, A154333(x) > 49.
%e A179386 For numbers x > 136, A154333(x) > 60.
%e A179386 For numbers x > 568, A154333(x) > 63.
%e A179386 For numbers x > 5215, A154333(x) > 174.
%e A179386 For numbers x > 367806, A154333(x) > 207.
%e A179386 For numbers x > 939787, A154333(x) > 307.
%t A179386 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; min = 10^100; Do[m = Floor[(n^3)^(1/2)]; k = n^3 - m^2; If[k != 0, If[k <= min, ile = 0; Do[If[vecd[[z]] < k, ile = ile + 1], {z, 1, len}]; len = ile + 1; min = 10^100; vecd[[len]] = k; vecx[[len]] = n; vecy[[len]] = m]], {n, 1, 13333677}]; dd = {}; xx = {}; yy = {}; Do[AppendTo[dd, vecd[[n]]]; AppendTo[xx, vecx[[n]]]; AppendTo[yy, vecy[[n]]], {n, 1, len}]; dd
%Y A179386 Cf. A179107, A179108, A179109, A179387, A179388.
%K A179386 more,nonn,hard
%O A179386 1,1
%A A179386 _Artur Jasinski_, Jul 13 2010, Aug 03 2010
%E A179386 Edited by _M. F. Hasler_, Sep 27 2013