%I #42 Feb 03 2023 01:32:21
%S 1,5,6,9,14,21,29,30,41,45,46,49,54,61,69,70,81,86,89,94,101,105,109,
%T 126,129,134,141,145,149,161,166,174,181,189,201,205,206,214,229,230,
%U 241,245,246,249,254,261,269,270,281,294,301,305,309,321,326,329,334,345,349,366,369,381,389,401,405
%N Positive integers k properly represented by the positive definite binary quadratic form X^2 + 5*Y^2 = k, in increasing order.
%C This is one of the bisections of sequence A343238. The other sequence is A344232.
%C This is a proper subsequence of A020669.
%C The primes in this sequence are given in A033205.
%C Discriminant Disc = -20 = -4*5. Class number h(-20) = A000003(5) = 2. The reduced primitive forms representing the two proper (determinant = +1) equivalence classes are the present principal form F1 = [1, 0, 5] and F2 = [2, 2, 3] treated in A344232.
%C A positive integer k is properly represented by some primitive form of Disc = -20 if and only if the congruence s^2 + 20 == 0 (mod 4*k) has a solution. See, e.g., Buell Proposition 41, p. 50, or Scholz-Schoeneberg Satz 74, p. 105. That is, x^2 + 5 == 0 (mod k), with s = 2*x. For the representative solutions x from {0, 1, ..., k-1}, with k from A343238, see A343239. These solutions x determine the so-called representative parallel primitive forms (rpapfs) [k, 2*x, (x^2 + 5)/k] representing k. They are properly equivalent (via so called R(t)-transformations) to one of the reduced forms F1 or F2. (See also W. Lang's links in A225953 and A324251, but there indefinite forms are considered.)
%C In order to find out which k from A343238 is represented either by form F1 or F2 the two generic multiplicative characters of Disc = -20, namely Legendre(k|p), with the odd prime p = 5 which divides Disc = -20, and Jacobi(-1|k) can be used. See Buell, pp. 51-52. They lead to the two classes of genera of Disc -20.
%C The present genus I, the principal one, has for odd primes p, not 5, the values Legendre(p|5) = Legendre(5|p) = +1 and Jacobi(-1|p) = Legendre(-1|p) = +1, leading for odd primes not equal to 5 to A033205. The prime 2 is not represented. The prime 5 is trivially represented. For the other genus II these two characters have values -1. There prime 2 is represented.
%C For composite k the prime number factorization is used, and for powers of primes the lifting theorem is employed (see, e.g., Apostol, p. 121, Theorem 5.30). The solution for prime 2 represented by form F2 = [2, 2, 3] (from the other genus II) is not liftable to powers of 2. The solution for prime 5 is also not liftable (proof by induction). The solutions of the other primes from A033205 and A106865 are uniquely liftable to powers of these primes. See A343238 for all properly represented k for Disc = -20.
%C For the present genus I the properly represented integers k 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 = 0 giving a(1) = 1. The odd primes pI_j are from A033205 (== {1, 9} (mod 20)), the primes pII_k are from the odd primes of A106865 (== {3, 7}(mod 20)). The exponents of the second product are restricted: if a = 1 then PII >= 1 and Sum_{k=1..PII} eII(k) is odd. If a = 0 then PII >= 0, and if PII >= 1 then this sum is even.
%C Neighboring numbers k (twins) begin: [5, 6], [29, 30], [45, 46], [69, 70], [205, 206], [229, 230], [245, 246], [269, 270], [405, 406], ...
%C For the solutions (X, Y) of F2 = [1, 0, 5] properly representing k = a(n) see A344233.
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp 121 - 122.
%D D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%D A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Göschen Band 5131, Walter de Gruyter, 1973.
%Y Cf. A000003, A020669, A033205, A106865, A225953, A324251, A343238, A343239, A343240, A344232, A344233.
%K nonn,easy
%O 1,2
%A _Wolfdieter Lang_, Jun 10 2021