cp's OEIS Frontend

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.

%I A344232 #30 Aug 07 2021 21:53:18
%S A344232 2,3,7,10,15,18,23,27,35,42,43,47,58,63,67,82,83,87,90,98,103,107,115,
%T A344232 122,123,127,135,138,147,162,163,167,178,183,202,203,207,210,215,218,
%U A344232 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
%N 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.
%C A344232 This is one of the bisections of sequence A343238. The other sequence is A344231.
%C A344232 This is a proper subsequence of A029718.
%C A344232 The primes in this sequence are given in A106865.
%C A344232 See A344231 for more details.
%C A344232 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.
%C A344232 The product of any two odd a(n), not divisible by 5, is congruent to {1,5} (mod 8). See Buell, 4),  p. 51.
%C A344232 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.
%C A344232 The neighboring numbers k (twins) begin: [42, 43], [82, 83], [122, 123] [162, 163], [202, 203], [282, 283], ...
%C A344232 For the solutions (X, Y) of F2 = [2, 2, 3] properly representing k = a(n) see A344234.
%D A344232 D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%D A344232 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Göschen Band 5131, Walter de Gruyter, 1973.
%Y A344232 Cf. A029718, A033205, A106865, A343238, A343239, A343240, A344231, A344234.
%K A344232 nonn,easy
%O A344232 1,1
%A A344232 _Wolfdieter Lang_, Jun 10 2021