A254064 Positive integers whose square is expressible in exactly one way as x^2 + 6xy + y^2, with 0 < x < y.

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 102, 103, 105, 112, 113, 115, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 151, 153, 154, 155, 158, 164, 167

Offset

1,1

Comments

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

n is in the sequence iff A331671(n)=1. - Ray Chandler, Feb 26 2020

Links

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 750 terms from Colin Barker)

Example

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).

Mathematica

s[n_] := Solve[0 < x < y && n^2 == x^2 + 6 x y + y^2, {x, y}, Integers];
Reap[For[n = 1, n < 200, n++, If[Length[s[n]]==1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 13 2020 *)

Crossrefs

Cf. A084645, A232437, A248599, A254063, A331671.

Sequence in context: A353440 A198390 A118905 * A257224 A092433 A056203

Adjacent sequences: A254061 A254062 A254063 * A254065 A254066 A254067

Keyword

nonn

Author

Colin Barker, Jan 24 2015

Status

approved