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Search: id:A107628
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| A107628 |
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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. |
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+0
1
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| 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
(list; graph; listen)
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OFFSET
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1,15
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COMMENT
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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 (noe(AT)sspectra.com), May 07 2008
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REFERENCES
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See A106856.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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EXAMPLE
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a(15)=2 because the forms x^2+xy+4y^2 and 2x^2+xy+2y^2 have discriminant -15.
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MATHEMATICA
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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
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PROGRAM
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(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 */
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CROSSREFS
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Cf. A106856 (start of many quadratic forms).
Sequence in context: A109708 A035468 A051777 this_sequence A115296 A059048 A124645
Adjacent sequences: A107625 A107626 A107627 this_sequence A107629 A107630 A107631
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), May 18 2005, Apr 30 2008
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