A248599 -id:A248599 - cp's OEIS frontend

A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 93, 95, 97, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 134, 139, 140, 143, 146, 148, 151, 152, 154, 155, 156, 157, 158, 161

Offset: 1

Jean-Christophe Hervé, Nov 24 2013

nonn

Analog of A084645 for 120-degree angle triangles with integer sides.
Numbers with exactly one prime divisor of the form 6k+1 with multiplicity one.
Primitive elements of A050931.

			a(1) = 7 as 7^2 = 3^2 + 3*5 + 5^2.

Cf. A002476, A050931, A230780, A232436 (subsequence).

Cf. A084645, A232437, A248599, A254063, A254064.

r[k_] := Reduce[x>0 && y>0 && k^2 == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, False, rk[[0]] === Or && Length[rk] == 2, True, True, False];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230780(k)*A002476(p) for k, p > 0, ordered by increasing values.

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

Original entry on oeis.org

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

Colin Barker, Jan 24 2015

nonn

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

			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).

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

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

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 *)

A254063 Positive integers whose square is expressible in exactly one way as x^2 + 5xy + y^2, with 0 < x < y.

Original entry on oeis.org

5, 10, 15, 17, 20, 30, 34, 35, 37, 40, 41, 43, 45, 47, 51, 55, 59, 60, 65, 67, 68, 70, 74, 79, 80, 82, 83, 86, 89, 90, 94, 95, 101, 102, 105, 109, 110, 111, 115, 118, 119, 120, 123, 127, 129, 130, 131, 134, 135, 136, 140, 141, 145, 148, 151, 153, 155, 158

Offset: 1

Colin Barker, Jan 24 2015

nonn

			5 is in the sequence because the only solution to x^2 + 5xy + y^2 = 25 with 0 < x < y is (x,y) = (1,3).

Colin Barker, Table of n, a(n) for n = 1..750

Cf. A084645, A232437, A248599, A254064.

cp's OEIS Frontend

A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A254063 Positive integers whose square is expressible in exactly one way as x^2 + 5xy + y^2, with 0 < x < y.

Views

Author

Keywords

Examples

Links

Crossrefs