A332588 Let t_k denote the triangular number k*(k+1)/2. Suppose 0 < x < y < z are integers satisfying t_x + t_y = t_p, t_y + t_z = t_q, t_x + t_z = t_r, for integers p,q,r. Sort the triples [x,y,z] first by x, then by y. Sequence gives the values of x.

9, 14, 20, 23, 27, 35, 35, 41, 44, 51, 54, 54, 62, 65, 65, 66, 74, 76, 77, 77, 77, 83, 90, 104, 104, 105, 105, 105, 114, 119, 124, 131, 131, 135, 135, 152, 152, 161, 161, 165, 168, 170, 170, 174, 189, 189, 189, 209, 210, 210, 216, 228, 230, 230, 237, 245, 252, 252

Offset

1,1

Comments

Ulas gives a table assuming 0 < x < y < z < 1000. Because of the assumption z < 1000, only the entries with x < 46 can be relied upon (above this it is possible that there are gaps in the table).

Links

Giovanni Resta, Table of n, a(n) for n = 1..162

Ulas Maciej, A note on Sierpinski's problem related to triangular numbers, arXiv:0810.0222 [math.NT], 2008. See Table 1.

Ulas Maciej, A note on Sierpinski's problem related to triangular numbers, Colloq. Math. 117 (2009), no. 2, 165-173. See MR2550124. See Table 1.

Giovanni Resta, Table of x, y, z, p, q, r for x < 1000

Example

The initial values of x, y, z, p, q, r are:
   x    y    z    p    q    r
  --  ---  ---  --- ----  ---
   9   13   44   16   46   45
  14   51  104   53  116  105
  20   50  209   54  215  210
  23   30   90   38   95   93
  27  124  377  127  397  378
  35   65   86   74  108   93
  35  123  629  128  641  630
  41  119  285  126  309  288
  44  245  989  249 1019  990
  ...

Mathematica

L = {}; t[n_] := n (n + 1)/2; Do[ syp = Solve[t[x] + t[y] == t[p] && p > 0 && y > x , {p, y}, Integers]; If[syp =!= {}, Do[{y1, p1} = {y, p} /. e; s = Solve[ t[y1] + t[z] == t[q] && t[x] + t[z] == t[r]  && q > 0 && z > y1 && r > 0, {z, q, r}, Integers]; If[s =!= {}, L = Join[L, {x, y1, z, p1, q, r} /. s]], {e, syp}]], {x, 54}]; Sort[L][[All, 1]] (* Giovanni Resta, Mar 02 2020 *)

Crossrefs

Cf. A000217, A332589, A332590, A332591, A332592, A332593.

Sequence in context: A327896 A302056 A173792 * A034703 A006624 A184218

Adjacent sequences: A332585 A332586 A332587 * A332589 A332590 A332591

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2020

Extensions

Terms a(10) and beyond from Giovanni Resta, Mar 02 2020

Status

approved