OFFSET
1,15
COMMENTS
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
REFERENCES
See A106856.
LINKS
T. D. Noe, Table of n, a(n) for n=1..10000
EXAMPLE
a(15)=2 because the forms x^2 + xy + 4y^2 and 2x^2 + xy + 2y^2 have discriminant -15.
MATHEMATICA
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
PROG
(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 */
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, May 18 2005, Apr 30 2008
STATUS
approved