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%I A290500 #23 Dec 30 2019 14:24:12
%S A290500 1953125,3906250,5859375,7812500,11718750,13671875,15625000,17578125,
%T A290500 21484375,23437500,27343750,31250000,35156250,37109375,41015625,
%U A290500 42968750,44921875,46875000,52734375,54687500,60546875,62500000,64453125,70312500,74218750,82031250
%N A290500 Hypotenuses for which there exist exactly 9 distinct integer triangles.
%C A290500 Numbers whose square is decomposable in 9 different ways into the sum of two nonzero squares: these are those with only one prime divisor of the form 4k+1 with multiplicity nine.
%H A290500 Ray Chandler, <a href="/A290500/b290500.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Hamdi Sahloul)
%F A290500 Terms are obtained by the product A004144(k)*A002144(p)^9 for k, p > 0 ordered by increasing values.
%e A290500 a(1) = 1953125 = 5^9, a(5) = 11718750 = 2*3*5^9, a(101) = 375000000 = 2^6*3*5^9.
%t A290500 r[a_]:={b, c}/.{ToRules[Reduce[0<b<c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[Range[375000000], Length[r[#]] == 9 &] (* _Vincenzo Librandi_, Mar 01 2016 *)
%Y A290500 Cf. A002144, A006339, A046080, A046109, A083025.
%Y A290500 Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).
%K A290500 nonn
%O A290500 1,1
%A A290500 _Hamdi Sahloul_, Aug 04 2017

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