A176835 - OEIS

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

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OFFSET

1,3

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

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A176836, A176837.

Sequence in context: A326034 A332207 A339929 * A353865 A237110 A078704

Adjacent sequences: A176832 A176833 A176834 * A176836 A176837 A176838

KEYWORD

nonn

AUTHOR

Jens Voß, Apr 27 2010

EXTENSIONS

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

STATUS

approved