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%I #15 Feb 26 2020 10:42:13
%S 7,14,17,21,23,28,31,34,35,41,42,46,47,51,56,62,63,68,69,70,71,73,77,
%T 79,82,84,85,89,91,92,93,94,97,102,103,105,112,113,115,123,124,126,
%U 127,133,136,137,138,140,141,142,146,151,153,154,155,158,164,167
%N Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.
%C Equivalently positive integers whose square is expressible in exactly one way as -x^2 + 2xy + y^2 with 0 < x < y by replacing (x,y) with (2x,x+y). As such this sequence represents the sum of legs that are unique to a single Pythagorean triangle. - _Ray Chandler_, Feb 18 2020
%C n is in the sequence iff A331671(n)=1. - _Ray Chandler_, Feb 26 2020
%H Ray Chandler, <a href="/A254064/b254064.txt">Table of n, a(n) for n = 1..10000</a> (first 750 terms from Colin Barker)
%e 7 is in the sequence because the only solution to x^2 + 6xy + y^2 = 49 with 0 < x < y is (x,y) = (2,3).
%t s[n_] := Solve[0 < x < y && n^2 == x^2 + 6 x y + y^2, {x, y}, Integers];
%t Reap[For[n = 1, n < 200, n++, If[Length[s[n]]==1, Print[n]; Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Feb 13 2020 *)
%Y Cf. A084645, A232437, A248599, A254063, A331671.
%K nonn
%O 1,1
%A _Colin Barker_, Jan 24 2015
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