A332591 - OEIS

%I #17 Mar 02 2020 20:09:23

%S 16,53,54,38,127,74,128,126,249,86,153,250,117,150,431,206,465,118,

%T 147,147,432,296,685,686,686,200,378,561,444,1023,1552,218,278,205,

%U 1024,621,1457,306,1866,431,603,499,1458,1022,359,1289,1999,2000,1065,1589,446,486

%N 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 p.

%C 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).

%H Giovanni Resta, <a href="/A332591/b332591.txt">Table of n, a(n) for n = 1..162</a>

%H Ulas Maciej, <a href="https://arxiv.org/abs/0810.0222">A note on Sierpinski's problem related to triangular numbers</a>, arXiv:0810.0222 [math.NT], 2008. See Table 1.

%H Ulas Maciej, <a href="http://dx.doi.org/10.4064/cm117-2-2">A note on Sierpinski's problem related to triangular numbers</a>, Colloq. Math. 117 (2009), no. 2, 165-173. See MR2550124. See Table 1.

%e The initial values of x, y, z, p, q, r are:

%e    x    y    z    p    q    r

%e   --  ---  ---  --- ----  ---

%e    9   13   44   16   46   45

%e   14   51  104   53  116  105

%e   20   50  209   54  215  210

%e   23   30   90   38   95   93

%e   27  124  377  127  397  378

%e   35   65   86   74  108   93

%e   35  123  629  128  641  630

%e   41  119  285  126  309  288

%e   44  245  989  249 1019  990

%e   ...

%t 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, 4]] (* _Giovanni Resta_, Mar 02 2020 *)

%Y Cf. A000217, A332588-A332593.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Feb 29 2020

%E Terms a(10) and beyond from _Giovanni Resta_, Mar 02 2020