

A107628


Number of integral quadratic forms ax^2 + bxy + cy^2 whose discriminant b^24ac is n, 0 <= b <= a <= c and gcd(a,b,c) = 1.


3



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, 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, 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
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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 2k1 (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^24a*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

Cf. A106856 (start of many quadratic forms).
Cf. A133675 (n such that a(n)=1).
Cf. A223708 (without zeros).
Sequence in context: A263860 A051777 A262709 * A268389 A288969 A305355
Adjacent sequences: A107625 A107626 A107627 * A107629 A107630 A107631


KEYWORD

nonn


AUTHOR

T. D. Noe, May 18 2005, Apr 30 2008


STATUS

approved



