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 A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)). 9
 4, 5, 7, 9, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 169, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. 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. LINKS Jianing Song, Table of n, a(n) for n = 1..10000 EXAMPLE norm((1 + sqrt(-19))/2) = norm((1 - sqrt(-19))/2) = 5; norm((3 + sqrt(-19))/2) = norm((3 - sqrt(-19))/2) = 7; norm((5 + sqrt(-19))/2) = norm((5 - sqrt(-19))/2) = 11; norm((7 + sqrt(-19))/2) = norm((7 - sqrt(-19))/2) = 17. PROG (PARI) isA341787(n) = my(disc=-19); (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1) CROSSREFS Cf. A011585, A106863, A191019, A191063. The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171. The total number of elements with norm n is given by A028641. 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. Sequence in context: A283554 A035243 A035255 * A335984 A286050 A047493 Adjacent sequences:  A341784 A341785 A341786 * A341788 A341789 A341790 KEYWORD nonn,easy AUTHOR Jianing Song, Feb 19 2021 STATUS approved

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Last modified August 3 13:59 EDT 2021. Contains 346438 sequences. (Running on oeis4.)