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).

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, 1343, 54, 84, 14, 185, 95, 65, 15, 41, 35, 42, 560, 9, 23, 140, 287, 24, 17, 39, 105, 1539, 10, 48, 18, 87, 1770, 104, 183, 216, 27, 455, 11, 200, 119, 45, 20, 71, 63, 72, 14060, 99

Offset: 2

Author

Carmine Suriano, Oct 14 2009

Keywords

• nonn

Comments

From R. J. Mathar, Oct 23 2010: (Start)

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:

n Seq # solutions

-- ------- ------------------------------------------------

2 A001652 3, 20, 119, 696, 4059

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

4 ...

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

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

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

8 A336625 15, 32, 527, 1104, 17919

9 ...

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

11 11, 21, 230, 429, 4598, 8568

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

13 12, 77, 845, 1494, 16302

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

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

16 ...

17 51, 84, 3399, 5576

18 27, 44, 935, 1512, 31779

19 19, 285, 455, 6649

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

(End) [table reformatted by Jon E. Schoenfield, Apr 01 2018]

Examples

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

Crossrefs

Cf. A000037.

Cf. A166478 (associated t). - R. J. Mathar, Oct 23 2010

Extensions

Deleted an 8 between 14 and 185. - R. J. Mathar, Oct 23 2010