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A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

%I A341787 #12 Feb 20 2021 07:56:28
%S A341787 4,5,7,9,11,17,19,23,43,47,61,73,83,101,131,137,139,149,157,163,169,
%T A341787 191,197,199,229,233,239,251,263,271,277,283,311,313,347,349,353,359,
%U A341787 367,389,397,419,443,457,461,463,467,479,491,499,503,541,557,571
%N A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).
%C A341787 Also norms of prime ideals in Z[(1+sqrt(-19))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
%C A341787 Consists of the primes such that (p,19) >= 0 and the squares of primes such that (p,19) = -1, where (p,19) is the Legendre symbol.
%C A341787 For primes p such that (p,19) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-19))/2], namely (x + y*(1+sqrt(-19))/2) and (x + y*(1-sqrt(-19))/2), where (x,y) is a solution to x^2 + x*y + 5*y^2 = p; for p = 19, (sqrt(-19)) is the unique ideal with norm p; for primes p with (p,19) = -1, (p) is the only ideal with norm p^2.
%H A341787 Jianing Song, <a href="/A341787/b341787.txt">Table of n, a(n) for n = 1..10000</a>
%e A341787 norm((1 + sqrt(-19))/2) = norm((1 - sqrt(-19))/2) = 5;
%e A341787 norm((3 + sqrt(-19))/2) = norm((3 - sqrt(-19))/2) = 7;
%e A341787 norm((5 + sqrt(-19))/2) = norm((5 - sqrt(-19))/2) = 11;
%e A341787 norm((7 + sqrt(-19))/2) = norm((7 - sqrt(-19))/2) = 17.
%o A341787 (PARI) isA341787(n) = my(disc=-19); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341787 Cf. A011585, A106863, A191019, A191063.
%Y A341787 The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.
%Y A341787 The total number of elements with norm n is given by A028641.
%Y A341787 Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), this sequence (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
%K A341787 nonn,easy
%O A341787 1,1
%A A341787 _Jianing Song_, Feb 19 2021