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A290502 - OEIS

A290502

Hypotenuses for which there exist exactly 14 distinct integer triangles.

24

6103515625, 12207031250, 18310546875, 24414062500, 36621093750, 42724609375, 48828125000, 54931640625, 67138671875, 73242187500, 85449218750, 97656250000, 109863281250, 115966796875, 128173828125, 134277343750, 140380859375, 146484375000, 164794921875

COMMENTS

Numbers whose square is decomposable in 14 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity fourteen.

FORMULA

Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.

EXAMPLE

a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.

MATHEMATICA

r[a_]:={b, c}/.{ToRules[Reduce[0<b<c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[Range[1171875000000], Length[r[#]] == 14 &] (* Vincenzo Librandi, Mar 01 2016 *)

CROSSREFS

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).