A366862 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.

1, 4, 9, 16, 12, 8, 25, 24, 31, 28, 36, 57, 33, 40, 49, 60, 127, 64, 52, 97, 121, 81, 84, 217, 72, 73, 100, 169, 96, 88, 7, 112, 105, 172, 3, 108, 48, 241, 177, 144, 199, 71, 145, 136, 129, 156, 124, 148, 180, 196, 161, 15, 184, 204, 225, 220, 391, 168, 372, 17, 44, 337, 385

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..63.

Wendy Appleby, Find all triples..., Number Theory group on LinkedIn.com, Oct 25 2023

Example

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.
a(2) = 4 is in the sequence since (a, b, c) = (28, 9, 4) is a solution.
a(3) = 9 is in the sequence since (a, b, c) = (52, 16, 9) is a solution.
a(4) = 16 is in the sequence since (a, b, c) = (57, 49, 16) is a solution.
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.
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.]
(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), ...

Prog

(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", "))))

Crossrefs

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

Sequence in context: A368103 A054580 A165488 * A369133 A368802 A313300

Adjacent sequences: A366859 A366860 A366861 * A366863 A366864 A366865

Keyword

nonn

Author

M. F. Hasler, Oct 25 2023

Status

approved