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A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

%I A084645 #47 Feb 16 2025 08:32:49
%S A084645 5,10,13,15,17,20,26,29,30,34,35,37,39,40,41,45,51,52,53,55,58,60,61,
%T A084645 68,70,73,74,78,80,82,87,89,90,91,95,97,101,102,104,105,106,109,110,
%U A084645 111,113,115,116,117,119,120,122,123,135,136,137,140,143,146,148,149
%N A084645 Hypotenuses for which there exists a unique integer-sided right triangle.
%C A084645 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 A084645 Ray Chandler, <a href="/A084645/b084645.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Zak Seidov)
%H A084645 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A084645 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 A084645 A046080(a(n)) = 1, A046109(a(n)) = 12. - _Jean-Christophe Hervé_, Dec 01 2013
%t A084645 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 A084645 (PARI) is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ _Hugo Pfoertner_, Sep 28 2024
%Y A084645 Cf. A002144, A006339, A046080, A046109, A083025.
%Y A084645 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 A084645 nonn
%O A084645 1,1
%A A084645 _Eric W. Weisstein_, Jun 01 2003