A254064 -id: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.

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.

A331671 Number of Pythagorean triangles with sum of legs n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1

Offset: 1

Ray Chandler, Feb 26 2020

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A118905, A198390, A254064.

A248599 Positive integers whose square is expressible in exactly one way as x^2 + 4xy + y^2, with 0 < x < y.

Original entry on oeis.org

11, 13, 22, 23, 26, 33, 37, 39, 44, 46, 47, 52, 55, 59, 61, 65, 66, 69, 71, 73, 74, 77, 78, 83, 88, 91, 92, 94, 97, 99, 104, 107, 109, 110, 111, 115, 117, 118, 122, 130, 131, 132, 138, 141, 142, 146, 148, 154, 156, 157, 161, 165, 166, 167, 176, 177, 179, 181

Offset: 1

Colin Barker, Jan 21 2015

nonn

			11 is in the sequence because the only solution to x^2 + 4xy + y^2 = 121 with 0 < x < y is (x,y) = (4,5).

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

Cf. A232437, A084645, A254063, A254064.

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.

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A331671 Number of Pythagorean triangles with sum of legs n.

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A248599 Positive integers whose square is expressible in exactly one way as x^2 + 4xy + y^2, with 0 < x < y.

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

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