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 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. 6
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified May 18 04:22 EDT 2021. Contains 343994 sequences. (Running on oeis4.)