%I #46 Sep 28 2024 21:11:40

%S 5,10,13,15,17,20,26,29,30,34,35,37,39,40,41,45,51,52,53,55,58,60,61,

%T 68,70,73,74,78,80,82,87,89,90,91,95,97,101,102,104,105,106,109,110,

%U 111,113,115,116,117,119,120,122,123,135,136,137,140,143,146,148,149

%N Hypotenuses for which there exists a unique integer-sided right triangle.

%C Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - _Jean-Christophe Hervé_, Nov 11 2013

%H Ray Chandler, <a href="/A084645/b084645.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Zak Seidov)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>

%F Terms are obtained by the products A004144(k)*A002144(p) for k, p > 0, ordered by increasing values. - _Jean-Christophe Hervé_, Nov 12 2013

%F A046080(a(n)) = 1, A046109(a(n)) = 12. - _Jean-Christophe Hervé_, Dec 01 2013

%t r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* _Jean-François Alcover_, Oct 22 2012 *)

%o (PARI) is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ _Hugo Pfoertner_, Sep 28 2024

%Y Cf. A002144, A006339, A046080, A046109, A083025.

%Y Cf. A004144 (0), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), 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 nonn

%O 1,1

%A _Eric W. Weisstein_, Jun 01 2003