%I #19 Oct 07 2022 10:18:52

%S 3,2,5,3,6,15,4,11,8,12,20,5,51,27,19,15,6,11,45,95,12,54,7,29,24,30,

%T 1343,54,84,14,185,95,65,15,41,35,42,560,9,23,140,287,24,17,39,105,

%U 1539,10,48,18,87,1770,104,183,216,27,455,11,200,119,45,20,71,63,72,14060,99

%N Minimum positive integer solution x of equation n=x*(x+1)/(t*(t+1)); that is, ratio of product of two consecutive integers divided by product of two consecutive integers. Here n is a nonsquare integer (see A000037).

%C From _R. J. Mathar_, Oct 23 2010: (Start)

%C Writing x = (-1 + sqrt(1 + 4*n*t*(t+1))/2, each solution is associated with a Diophantine equation 1 + 4*n*t*(t+1) = s^2. The sequence entries are the leading column if all solutions are presented in rows for a given n:

%C n Seq # solutions

%C -- ------- ------------------------------------------------

%C 2 A001652 3, 20, 119, 696, 4059

%C 3 A001571 2, 9, 35, 132, 494, 1845, 6887

%C 4 ...

%C 5 A077262 5, 14, 99, 260, 1785, 4674

%C 6 A077291 3, 8, 35, 84, 351, 836, 3479, 8280

%C 7 A077401 6, 14, 104, 231, 1665, 3689

%C 8 A336625 15, 32, 527, 1104, 17919

%C 9 ...

%C 10 A341895 4, 20, 39, 175, 779, 1500, 6664, 29600

%C 11 11, 21, 230, 429, 4598, 8568

%C 12 8, 15, 119, 216, 1664, 3015, 23183

%C 13 12, 77, 845, 1494, 16302

%C 14 20, 35, 615, 1064, 18444, 31899

%C 15 5, 9, 44, 75, 350, 594, 2759, 4680, 21725, 36849

%C 16 ...

%C 17 51, 84, 3399, 5576

%C 18 27, 44, 935, 1512, 31779

%C 19 19, 285, 455, 6649

%C 20 15, 24, 279, 440, 5015, 7904

%C (End) [table reformatted by _Jon E. Schoenfield_, Apr 01 2018]

%e For n=14, x=20; corresponding value of t is 5 since 14 = 20*21/(5*6).

%Y Cf. A000037.

%Y Cf. A166478 (associated t). - _R. J. Mathar_, Oct 23 2010

%K nonn

%O 2,1

%A _Carmine Suriano_, Oct 14 2009

%E Deleted an 8 between 14 and 185. - _R. J. Mathar_, Oct 23 2010