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%I #19 Dec 30 2019 14:34:27
%S 6103515625,12207031250,18310546875,24414062500,36621093750,
%T 42724609375,48828125000,54931640625,67138671875,73242187500,
%U 85449218750,97656250000,109863281250,115966796875,128173828125,134277343750,140380859375,146484375000,164794921875
%N Hypotenuses for which there exist exactly 14 distinct integer triangles.
%C 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.
%H Ray Chandler, <a href="/A290502/b290502.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Hamdi Sahloul)
%F Terms are obtained by the product A004144(k)*A002144(p)^14 for k, p > 0 ordered by increasing values.
%e a(1) = 6103515625 = 5^14, a(5) = 36621093750 = 2*3*5^14, a(101) = 1171875000000 = 2^6*3*5^14.
%t 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 *)
%Y Cf. A002144, A006339, A046080, A046109, A083025.
%Y 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).
%K nonn
%O 1,1
%A _Hamdi Sahloul_, Aug 04 2017