A344232 All positive integers k properly represented by the positive definite binary quadratic form 2*X^2 + 2*X*Y + 3*Y^2 = k, in increasing order.

2, 3, 7, 10, 15, 18, 23, 27, 35, 42, 43, 47, 58, 63, 67, 82, 83, 87, 90, 98, 103, 107, 115, 122, 123, 127, 135, 138, 147, 162, 163, 167, 178, 183, 202, 203, 207, 210, 215, 218, 223, 227, 235, 243, 258, 263, 267, 282, 283, 287, 290, 298, 303, 307, 315, 322, 327, 335, 343, 347, 362, 367, 378, 383, 387

Offset: 1

Author

Wolfdieter Lang, Jun 10 2021

Keywords

• nonn
• easy

Comments

This is one of the bisections of sequence A343238. The other sequence is A344231.

This is a proper subsequence of A029718.

The primes in this sequence are given in A106865.

See A344231 for more details.

The reduced form [2, 2, 3] represents the proper (determinant +1) equivalence class of one of the two genera (genus II) of discriminant -20. The multiplicative generic characters for discriminant Disc = -20 have values Jacobi(a(n)|5) = -1 and Jacobi(-1|a(n)) = -1, for odd a(n) not divisible by 5. See Buell, p. 52.

The product of any two odd a(n), not divisible by 5, is congruent to {1,5} (mod 8). See Buell, 4), p. 51.

For this genus II of Disc = -20 the positive integers represented are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII}(pII_k)^(eII(k)), with a and b from {0, 1}, but if PI = PII = 0 (empty products are 1) then (a, b) = (1, 0) or (1, 1), giving a(1) = 2 or a(4) = 10. The odd primes pI_j are from A033205 and the odd primes pII_j from the odd primes of A106865. The exponents of the second product satisfy: if a = 1 then PII >= 0, and if PII >=1 then Sum_{k=1..PII} eII(j) is even. If a = 0 then PII >= 1 and this sum is odd.

The neighboring numbers k (twins) begin: [42, 43], [82, 83], [122, 123] [162, 163], [202, 203], [282, 283], ...

For the solutions (X, Y) of F2 = [2, 2, 3] properly representing k = a(n) see A344234.

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Göschen Band 5131, Walter de Gruyter, 1973.

Crossrefs

Cf. A029718, A033205, A106865, A343238, A343239, A343240, A344231, A344234.