A176835 Number of positive integers k for which k^2 - n*k is a square.

1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 5, 3, 2, 3, 2, 2, 5, 2, 2, 5, 3, 2, 4, 2, 2, 5, 2, 4, 5, 2, 5, 3, 2, 2, 5, 5, 2, 5, 2, 2, 8, 2, 2, 8, 3, 3, 5, 2, 2, 4, 5, 5, 5, 2, 2, 5, 2, 2, 8, 5, 5, 5, 2, 2, 5, 5, 2, 8, 2, 2, 8, 2, 5, 5, 2, 8, 5, 2, 2, 5, 5, 2, 5, 5, 2, 8, 5, 2, 5, 2, 5, 11, 2, 3, 8, 3

Offset: 1

Author

Jens Voß, Apr 27 2010

Keywords

• nonn

Comments

The number of solutions to the Diophantine equation x^2 - 4*y^2 = n^2 for x,y >= 0, where k = (1+x)/2 and k^2 - n*k = y^2. - R. J. Mathar, May 02 2010

Examples

			A176835(9) = 3 because the three numbers 9^2 - 9*9 = 0, 12^2 - 9*12 = 36 and 25^2 - 9*25 = 400 are squares.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A176836, A176837.

Programs

Maple

A176835 := proc(n) N := n^2 ; a := 0 ; for d in numtheory[divisors](N) do d2 := N/d ; x := (d+d2)/2 ; y := (d-d2)/4 ; if type(x,'integer') and type(y,'integer') and x>=0 and y >= 0 then a := a+1 ; end if; end do: a ; end proc: seq(A176835(n),n=1..100) ; # R. J. Mathar, May 02 2010

Mathematica

a[n_] := Solve[k >= n && x >= 0 && k^2 - n k == x^2, {k, x}, Integers] // Length;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 20 2023 *)

PARI

A176835(n) = { my(n2 = n^2, s=0); fordiv(n2,d,my(d2=n2/d, x = (d+d2)/2, y = (d-d2)/4); if((1==denominator(x))&&(1==denominator(y))&&(x>=0)&&(y>=0),s++)); (s); }; \\ Antti Karttunen, Sep 28 2018, after R. J. Mathar's Maple-program

Formula

a(n) = A176836(n) - A176837(n).

Extensions

More terms from R. J. Mathar, May 02 2010