cp's OEIS Frontend

A244819 Positive numbers primitively represented by the binary quadratic form (1, 0, 3).

%I A244819 #13 Mar 07 2021 17:38:01
%S A244819 1,3,4,7,12,13,19,21,28,31,37,39,43,49,52,57,61,67,73,76,79,84,91,93,
%T A244819 97,103,109,111,124,127,129,133,139,147,148,151,156,157,163,169,172,
%U A244819 181,183,193,196,199,201,211,217,219,223,228,229,237,241,244,247,259
%N A244819 Positive numbers primitively represented by the binary quadratic form (1, 0, 3).
%C A244819 Discriminant = -12.
%C A244819 x^2 + 3*y^2 represents positive k properly (gcd(x, y) = 1), with nonnegative x, and the following multiplicities m(k): m(1) = 1, m(3) = 1, m(4) = 2, and if k = 3^a*4^b*Product_{j=1..P1} p1(j)^e1(j), with p1(j) primes 1 (mod 6) (A002476), e1(j) nonnegative integer numbers, and a and b from {0, 1}, then m(k) = 2^(P1+b). Shown by the lifting theorem (e.g., Apostol) for prime powers. Note that for prime 2 there is one solution of j^2 + 3 == 0 (mod 2) this corresponds the imprimitive reduced form (2, 2, 2), not to the one reduced primitive form (1, 0, 3) for discriminant -12 (A000003(3) = 1). - _Wolfdieter Lang_, Mar 02 2021
%D A244819 Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976 (1986), Theorem 5.30, pp. 121-122.
%H A244819 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%e A244819 Proper solution to x^2 + 3*y^2 = a(n), with x nonnegative: a(12 = 3*4) with (x, y) = (3, pm 1), pm = +1 or -1, multiplicity m(12) = 2, (a, b, P1) = (1, 1, 0); a(21 = 3*7) with (3, pm 2), m(21) = 2, (a, b, P1) = (1, 0, 1); a(49 = 7^2) with (1, pm 4), m(49) = 2 (a, b, P1) = (0, 0, 1)). - _Wolfdieter Lang_, Mar 02 2021
%p A244819 # Function PriRepBQF in A244779.
%p A244819 A244819 list := n -> PriRepBQF(1, 0, 3, n); A244819_list(259);
%t A244819 Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + 3 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* _Jean-François Alcover_, Oct 31 2016 *)
%Y A244819 Cf. A002476, A092574, A244779, A244780.
%K A244819 nonn
%O A244819 1,2
%A A244819 _Peter Luschny_, Jul 06 2014