cp's OEIS Frontend

A166477 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).

%I A166477 #22 Feb 05 2025 05:49:27
%S A166477 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 A166477 1343,54,84,14,185,95,65,15,41,35,42,560,9,23,140,287,24,17,39,105,
%U A166477 1539,10,48,18,87,1770,104,183,216,27,455,11,200,119,45,20,71,63,72,14060,99
%N A166477 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 A166477 From _R. J. Mathar_, Oct 23 2010: (Start)
%C A166477 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 A166477    n  Seq #  solutions
%C A166477   -- ------- ------------------------------------------------
%C A166477    2 A001652 3, 20, 119, 696, 4059
%C A166477    3 A001571 2, 9, 35, 132, 494, 1845, 6887
%C A166477    4         ...
%C A166477    5 A077262 5, 14, 99, 260, 1785, 4674
%C A166477    6 A077291 3, 8, 35, 84, 351, 836, 3479, 8280
%C A166477    7 A077401 6, 14, 104, 231, 1665, 3689
%C A166477    8 A336625 15, 32, 527, 1104, 17919
%C A166477    9         ...
%C A166477   10 A341895 4, 20, 39, 175, 779, 1500, 6664, 29600
%C A166477   11         11, 21, 230, 429, 4598, 8568
%C A166477   12         8, 15, 119, 216, 1664, 3015, 23183
%C A166477   13         12, 77, 845, 1494, 16302
%C A166477   14         20, 35, 615, 1064, 18444, 31899
%C A166477   15         5, 9, 44, 75, 350, 594, 2759, 4680, 21725, 36849
%C A166477   16         ...
%C A166477   17         51, 84, 3399, 5576
%C A166477   18         27, 44, 935, 1512, 31779
%C A166477   19         19, 285, 455, 6649
%C A166477   20         15, 24, 279, 440, 5015, 7904
%C A166477 (End) [table reformatted by _Jon E. Schoenfield_, Apr 01 2018]
%e A166477 For n=14, x=20; corresponding value of t is 5 since 14 = 20*21/(5*6).
%Y A166477 Cf. A000037.
%Y A166477 Cf. A166478 (associated t). - _R. J. Mathar_, Oct 23 2010
%K A166477 nonn
%O A166477 2,1
%A A166477 _Carmine Suriano_, Oct 14 2009
%E A166477 Deleted an 8 between 14 and 185. - _R. J. Mathar_, Oct 23 2010