A166478 -id:A166478 - 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).

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

Carmine Suriano, Oct 14 2009

nonn

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
  -- ------- ------------------------------------------------
A001652 3, 20, 119, 696, 4059
A001571 2, 9, 35, 132, 494, 1845, 6887
       ...
A077262 5, 14, 99, 260, 1785, 4674
A077291 3, 8, 35, 84, 351, 836, 3479, 8280
A077401 6, 14, 104, 231, 1665, 3689
A336625 15, 32, 527, 1104, 17919
       ...
A341895 4, 20, 39, 175, 779, 1500, 6664, 29600
       11, 21, 230, 429, 4598, 8568
       8, 15, 119, 216, 1664, 3015, 23183
       12, 77, 845, 1494, 16302
       20, 35, 615, 1064, 18444, 31899
       5, 9, 44, 75, 350, 594, 2759, 4680, 21725, 36849
       ...
       51, 84, 3399, 5576
       27, 44, 935, 1512, 31779
       19, 285, 455, 6649
       15, 24, 279, 440, 5015, 7904
(End) [table reformatted by Jon E. Schoenfield, Apr 01 2018]

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

Cf. A000037.

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

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

A227054 a(n) = least triangular number t > 0 such that n*t is a triangular number, or 0 if no such t exists.

Original entry on oeis.org

1, 3, 1, 0, 3, 1, 3, 15, 0, 1, 6, 3, 6, 15, 1, 0, 78, 21, 10, 6, 1, 3, 45, 190, 0, 3, 55, 1, 15, 10, 15, 28203, 45, 105, 3, 1, 465, 120, 55, 3, 21, 15, 21, 3570, 1, 6, 210, 861, 0, 6, 3, 15, 105, 21945, 1, 21, 3, 66, 26565, 91, 276, 378, 6, 0, 1596, 1, 300

Offset: 1

Alex Ratushnyak, Jun 29 2013

nonn

a(n) = 1 if and only if n is a triangular number.
Indices of conjectured 0's: 4, 9, 16, 25, 49, 64, 81, 121, 144, 169, 225, 256, 289, 361, 400, 441, 529, 576, 625, 729, ... These are squares of 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27.
a(n) = 0 if n = p^(2*j) where p is a prime and j > 0. - Jon E. Schoenfield, Sep 17 2023

			a(614) = 13964154294535688630985 = A000217(167117648945) because 614 * a(614) = A000217(4141012131555), and none of the smaller triangular numbers t satisfies 614*t = A000217(m) for some m.

Cf. A166478 (indices of t in A000217), A225502, A225503.

a(1)-a(25) and a(49)-a(67) from Jon E. Schoenfield, Sep 17 2023

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).

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A227054 a(n) = least triangular number t > 0 such that n*t is a triangular number, or 0 if no such t exists.

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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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A227054 a(n) = least triangular number t > 0 such that n*t is a triangular number, or 0 if no such t exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

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).