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A107628 Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1. 3

%I

%S 0,0,1,1,0,0,1,1,0,0,1,1,0,0,2,1,0,0,1,2,0,0,2,2,0,0,1,1,0,0,2,2,0,0,

%T 2,2,0,0,3,2,0,0,1,2,0,0,3,2,0,0,2,2,0,0,3,3,0,0,2,2,0,0,3,2,0,0,1,3,

%U 0,0,4,2,0,0,2,2,0,0,3,3,0,0,2,4,0,0,4,2,0,0,2,2,0,0,5,4,0,0,2,2,0,0,3,4,0

%N Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^2-4ac is -n, 0 <= b <= a <= c and gcd(a,b,c) = 1.

%C This sequence is closely related to the class number function, h(-n), which is given for fundamental discriminants in A006641. For a fundamental discriminant d, we have h(-d) < 2a(d). It appears that a(n) < Sqrt(n) for all n. For k>1, the primes p for which a(p)=k coincide with the numbers n such that the class number h(-n) is 2k-1 (see A006203, A046002, A046004, A046006. A046008, A046010, A046012, A046014, A046016 A046018, A046020). - _T. D. Noe_, May 07 2008

%D See A106856.

%H T. D. Noe, <a href="/A107628/b107628.txt">Table of n, a(n) for n=1..10000</a>

%e a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.

%t dLim=150; cnt=Table[0, {dLim}]; nn=Ceiling[dLim/4]; Do[d=b^2-4a*c; If[GCD[a, b, c]==1 && 0<-d<=dLim, cnt[[ -d]]++ ], {b, 0, nn}, {a, b, nn}, {c, a, nn}]; cnt

%o (PARI) {a(n)=local(m); if(n<3, 0, forvec(v=vector(3,k,[0,(n+1)\4]), if( (gcd(v)==1)&(-v[1]^2+4*v[2]*v[3]==n), m++ ), 1); m)} /* _Michael Somos_, May 31 2005 */

%Y Cf. A106856 (start of many quadratic forms).

%Y Cf. A133675 (n such that a(n)=1).

%Y Cf. A223708 (without zeros).

%K nonn

%O 1,15

%A _T. D. Noe_, May 18 2005, Apr 30 2008

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)