A366862 - OEIS

%I #14 Oct 30 2023 07:50:02

%S 1,4,9,16,12,8,25,24,31,28,36,57,33,40,49,60,127,64,52,97,121,81,84,

%T 217,72,73,100,169,96,88,7,112,105,172,3,108,48,241,177,144,199,71,

%U 145,136,129,156,124,148,180,196,161,15,184,204,225,220,391,168,372,17,44,337,385

%N Positive integers c such that there exist integers a > b > c such that ab+c, ac+b and bc+a are perfect squares, in order of increasing minimal a.

%C Are c = 2 and/or c = 5 in this sequence?

%H Wendy Appleby, <a href="https://www.linkedin.com/feed/update/urn:li:activity:7119351193918689280">Find all triples...</a>, Number Theory group on LinkedIn.com, Oct 25 2023

%e a(1) = 1 is in the sequence since (a, b, c) = (9, 7, 1) is a solution : 9*7 + 1 = 64, 9*1 + 7 = 16 = 7*1 + 9 are squares.

%e a(2) = 4 is in the sequence since (a, b, c) = (28, 9, 4) is a solution.

%e a(3) = 9 is in the sequence since (a, b, c) = (52, 16, 9) is a solution.

%e a(4) = 16 is in the sequence since (a, b, c) = (57, 49, 16) is a solution.

%e a(5) = 12 is in the sequence since (a, b, c) = (73, 24, 12) is a solution. It is listed after a(4) = 16 because the smallest a-value of a solution with c = 12 is 73, larger than the smallest possible a (= 57) for c = a(4) = 16.

%e The triples corresponding to the first terms of the sequence are: [We have highlighted some triples with ***, when the c-value is particularly small with respect to the a-value.]

%e (9, 7, 1), (28, 9, 4), (52, 16, 9), (57, 49, 16), (73, 24, 12), (89, 17, 8), (124, 36, 25), (129, 40, 24), (129, 97, 31), (168, 57, 28), (172, 49, 36), (177, 112, 57), (192, 64, 33), (201, 60, 40), (228, 64, 49), (289, 84, 60), (289, 161, 127), (292, 81, 64), (313, 108, 52), (339, 241, 97), (345, 280, 121), (364, 100, 81), (393, 112, 84), (408, 268, 217), (409, 136, 72), (432, 148, 73), (444, 121, 100), (456, 220, 169), (480, 145, 96), (504, 169, 88), (513, 9, 7)***, (513, 144, 112), (520, 156, 105), (568, 273, 172), (577, 33, 3)***, (577, 184, 108), (577, 528, 48), (579, 337, 241), (616, 529, 177), (628, 169, 144), (649, 449, 199), (649, 577, 71), (696, 204, 145), (705, 220, 136), (724, 240, 129), (744, 217, 156), (753, 264, 124), (784, 249, 148), (801, 220, 180), (844, 225, 196), (883, 721, 161), (889, 121, 15), (921, 280, 184), (952, 273, 204), (964, 256, 225), (969, 264, 220), (969, 577, 391), (984, 337, 168), (1012, 532, 372), (1016, 152, 17)***, (1016, 665, 44), (1059, 721, 337), (1065, 856, 385), (1092, 289, 256), (1104, 372, 193), (1129, 388, 192), (1153, 312, 264), (1212, 424, 201), (1228, 324, 289), (1236, 385, 240), (1240, 376, 249), (1240, 760, 441), (1248, 352, 273), (1312, 369, 288), (1321, 444, 232), (1345, 396, 280), (1353, 364, 312), (1360, 409, 276), (1372, 361, 324), (1377, 81, 19)***, (1393, 492, 228), (1428, 457, 268), (1459, 881, 577), (1524, 400, 361), (1524, 465, 304), (1537, 513, 63), (1569, 420, 364), (1569, 1057, 511), (1584, 441, 352), (1601, 1520, 80), (1648, 544, 297), (1656, 460, 369), (1684, 441, 400), (1764, 585, 316), (1780, 556, 345), (1788, 633, 292), (1800, 529, 376), (1801, 480, 420), (1801, 1351, 449), (1824, 624, 313), (1849, 532, 396), (1944, 568, 409), (1953, 193, 67), (1992, 673, 348), (2004, 697, 336), (2028, 529, 484), ...

%o (PARI) S=[]; for(a=1,oo, for(b=2,a-1, for(c=1,b-1, issquare(a*b+c)&& issquare(a*c+b)&& issquare(b*c+a)&& (S=setunion(S,[c])) &&print1(c", "))))

%Y Cf. A366865 (list of possible a-values).

%K nonn

%O 1,2

%A _M. F. Hasler_, Oct 25 2023