cp's OEIS Frontend

A006371 Number of positive definite reduced binary quadratic forms of discriminant -A014601(n).

%I A006371 M0207 #28 Jun 01 2025 10:04:06
%S A006371 1,1,1,1,1,2,2,2,1,2,3,2,2,2,3,3,2,3,4,2,1,4,5,4,2,2,4,4,3,4,5,4,1,4,
%T A006371 7,3,3,4,5,6,3,4,6,2,2,6,8,6,3,3,5,6,3,6,8,4,2,6,10,4,2,6,5,7,5,4,8,4,
%U A006371 3,8,10,8,3,2,7,6,4,8,10,6,1,8
%N A006371 Number of positive definite reduced binary quadratic forms of discriminant -A014601(n).
%D A006371 H. Davenport, The Higher Arithmetic. Cambridge Univ. Press, 5th edition, 1982, p. 144.
%D A006371 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006371 Robin Visser, <a href="/A006371/b006371.txt">Table of n, a(n) for n = 1..10000</a>
%H A006371 Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
%F A006371 a(2*n) = A006374(n) for all n > 0. - _Robin Visser_, May 29 2025
%e A006371 For n = 6, the a(6) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(6) = -12 are x^2 + 3*y^2 and 2*x^2 + 2*x*y + 2*y^2.  For n = 7, the a(7) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(7) = -15 are x^2 + x*y + 4*y^2 and 2*x^2 + x*y + 2*y^2. For n = 8, the a(8) = 2 positive definite reduced binary quadratic forms of discriminant -A014601(8) = -16 are x^2 + 4*y^2 and 2*x^2 + 2*y^2. - _Robin Visser_, May 29 2025
%o A006371 (SageMath)
%o A006371 def a(n):
%o A006371     D, ans = 2*n+1-(n+1)%2, 0
%o A006371     for b in range(-isqrt(D/3), isqrt(D/3)+1):
%o A006371         if ((D+b^2)%4 != 0): continue
%o A006371         for a in Integer((D+b^2)/4).divisors():
%o A006371             if ((abs(b)==a) or (a^2==(D+b^2)/4)) and (b < 0): continue
%o A006371             if (a >= abs(b)) and (a^2 <= (D+b^2)/4): ans += 1
%o A006371     return ans  # _Robin Visser_, May 29 2025
%Y A006371 Cf. A006374, A014601.
%K A006371 nonn
%O A006371 1,6
%A A006371 _N. J. A. Sloane_
%E A006371 More terms from _Sean A. Irvine_, Mar 19 2017
%E A006371 Name clarified and offset corrected by _Robin Visser_, May 29 2025