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 A364248 #62 Feb 16 2025 08:34:06 %S A364248 0,2,5,9,14,0,27,5,44,12,65,21,0,32,119,9,152,35,21,77,230,0,275,117, %T A364248 54,14,377,41,434,32,55,221,0,27,629,285,52,20,779,49,860,11,21,437, %U A364248 1034,0,1127,75,34,182,1325,27,110,159,19,725,1652,10,1769,837,0,320,195,99,2144,374 %N A364248 For n >= 3, a(n) is the least r >= 0 such that the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2 has an integer solution. %C A364248 The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section. %C A364248 For n >= 4, n*(n - 3)/2 >= a(n) >= 2. %H A364248 Robin Visser, <a href="/A364248/b364248.txt">Table of n, a(n) for n = 3..10000</a> %H A364248 Josef Gebel, Attila Pethő, and Horst G. Zimmer, <a href="https://doi.org/10.1007/3-540-61581-4_52">Computing S-integral points on elliptic curves</a>, Rend. Circ. Matem. Palermo 30 (1910), 377-407. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. %H A364248 The L-functions and modular forms database, <a href="https://www.lmfdb.org/">LMFDB</a> %H A364248 Joseph H. Silverman, <a href="https://www.math.ens.psl.eu/~benoist/refs/Silverman.pdf">The Arithmetic of Elliptic Curves</a> %H A364248 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a> %F A364248 a(p) = p*(p - 3)/2, p >= 5 a prime number. %F A364248 a(n) = 0 for n from A005563. %e A364248 n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47. Least r is 9 thus a(6) = 9 and [y, n, n+r] is [21, 6, 15]. %e A364248 n = 7: y^2 = 7^3 + (7 + r)^2 is valid for r = 14, 164. Least r is 14 thus a(7) = 14 and [y, n, n+r] is [28, 7, 21]. %o A364248 (PARI) a(n)=vecmin(((select((x)->x[1]>=0&&x[2]>=n,thue(thueinit(x^2-1,1),n^3)))~[1]))-n \\ _Thomas Scheuerle_, Sep 03 2023 %o A364248 (Sage) %o A364248 def a(n): %o A364248 for d in Integer(n^3).divisors(): %o A364248 if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): %o A364248 return (d-n^3/d)/2 - n # _Robin Visser_, Sep 30 2023 %Y A364248 Cf. A000040, A000096, A000217, A002061, A005563, A006254, A054504, A081119, A364421. %K A364248 nonn %O A364248 3,2 %A A364248 _Ctibor O. Zizka_, Sep 01 2023