A176835 -id:A176835 - cp's OEIS frontend

A176837 Number of pairs (a, b) of positive integers with a < b such that both a^2 - nb and b^2 - na are squares.

0, 0, 2, 1, 4, 2, 6, 5, 9, 4, 8, 13, 12, 6, 21, 18, 14, 9, 14, 28, 27

Offset: 1

Jens Voß, Apr 27 2010

nonn

			A176837(3) = 2 because the pairs (5, 8) and (7, 11) have the specified property: 5^2 - 3*8 = 1^2, 8^2 - 3*5 = 7^2; 7^2 - 3*11 = 4^2, 11^2 - 3*7 = 10^2.

A176837(n) = A176836(n) - A176835(n)

A176836 Number of pairs (a, b) of positive integers with a <= b such that both a^2 - nb and b^2 - na are squares.

Original entry on oeis.org

1, 1, 4, 2, 6, 4, 8, 7, 12, 6, 10, 15, 14, 8, 26, 21, 16, 12, 16, 30, 32

Offset: 1

Jens Voß, Apr 27 2010

nonn

			a(4) = 2 because the pairs (4, 4) and (5, 6) have the specified property: 4^4 - 4*4 = 0^2; 5^2 - 4*6 = 1^2, 6^2 - 4*5 = 4^2.

a(n) = A176835(n) + A176837(n)

A173636 Number of positive solutions of equation x(x+n)=y*y.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 2, 1, 2, 1, 1, 4, 1, 1, 4, 2, 1, 3, 1, 1, 4, 1, 3, 4, 1, 4, 2, 1, 1, 4, 4, 1, 4, 1, 1, 7, 1, 1, 7, 2, 2, 4, 1, 1, 3, 4, 4, 4, 1, 1, 4, 1, 1, 7, 4, 4, 4, 1, 1, 4, 4, 1, 7, 1, 1, 7, 1, 4, 4

Offset: 0

Carmine Suriano, Nov 23 2010

nonn

The solution x=y=0 is not counted.

Same as A115878 except for a(0). - Georg Fischer, Oct 12 2018

			a(9)=2 since x(x+9)=y*y has 2 solutions: x=3, y=6 and x=16, y=20.

Cf. A067721, A115878.

Join[{0}, Table[Length[{ToRules[Reduce[x (x + n) == y^2 && x > 0 && y > 0, {x, y}, Integers]]}], {n, 100}]]

a(n) = A176835(n)-1. - R. J. Mathar, Nov 23 2010

cp's OEIS Frontend

A176837 Number of pairs (a, b) of positive integers with a < b such that both a^2 - nb and b^2 - na are squares.

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A176836 Number of pairs (a, b) of positive integers with a <= b such that both a^2 - nb and b^2 - na are squares.

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A173636 Number of positive solutions of equation x(x+n)=y*y.

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