%I #16 Oct 20 2023 06:45:28
%S 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,
%T 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,
%U 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
%N Number of positive integers k for which k^2 - n*k is a square.
%C 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
%H Antti Karttunen, <a href="/A176835/b176835.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A176836(n) - A176837(n).
%e 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.
%p 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
%t a[n_] := Solve[k >= n && x >= 0 && k^2 - n k == x^2, {k, x}, Integers] // Length;
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 20 2023 *)
%o (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
%Y Cf. A176836, A176837.
%K nonn
%O 1,3
%A _Jens Voß_, Apr 27 2010
%E More terms from _R. J. Mathar_, May 02 2010