A341787 - OEIS

%I #12 Feb 20 2021 07:56:28

%S 4,5,7,9,11,17,19,23,43,47,61,73,83,101,131,137,139,149,157,163,169,

%T 191,197,199,229,233,239,251,263,271,277,283,311,313,347,349,353,359,

%U 367,389,397,419,443,457,461,463,467,479,491,499,503,541,557,571

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

%C 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 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 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 Jianing Song, <a href="/A341787/b341787.txt">Table of n, a(n) for n = 1..10000</a>

%e norm((1 + sqrt(-19))/2) = norm((1 - sqrt(-19))/2) = 5;

%e norm((3 + sqrt(-19))/2) = norm((3 - sqrt(-19))/2) = 7;

%e norm((5 + sqrt(-19))/2) = norm((5 - sqrt(-19))/2) = 11;

%e norm((7 + sqrt(-19))/2) = norm((7 - sqrt(-19))/2) = 17.

%o (PARI) isA341787(n) = my(disc=-19); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

%Y Cf. A011585, A106863, A191019, A191063.

%Y The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.

%Y The total number of elements with norm n is given by A028641.

%Y 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 nonn,easy

%O 1,1

%A _Jianing Song_, Feb 19 2021