A244779 -id:A244779 - cp's OEIS frontend

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

1, 3, 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 172, 181, 183, 193, 196, 199, 201, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247, 259

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -12.

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

			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

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976 (1986), Theorem 5.30, pp. 121-122.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A002476, A092574, A244779, A244780.

# Function PriRepBQF in A244779.
A244819 list := n -> PriRepBQF(1, 0, 3, n); A244819_list(259);

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 *)

A244780 Positive numbers primitively represented by the binary quadratic form (1, 1, 3).

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 23, 25, 27, 31, 33, 37, 45, 47, 53, 55, 59, 67, 69, 71, 75, 81, 89, 93, 97, 99, 103, 111, 113, 115, 125, 135, 137, 141, 155, 157, 159, 163, 165, 177, 179, 181, 185, 191, 199, 201, 207, 213, 223, 225, 229, 235, 243, 251, 253, 257, 265, 267

Offset: 1

Peter Luschny, Jul 06 2014

nonn

Discriminant = -11.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A244779, A244819.

# Function PriRepBQF in A244779.
A244780_list := n -> PriRepBQF(1, 1, 3, n); A244780_list(267);

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y + 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 *)

cp's OEIS Frontend

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

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