A296921 -id:A296921 - cp's OEIS frontend

A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).

4, 9, 25, 41, 43, 47, 49, 53, 61, 71, 83, 97, 113, 121, 131, 151, 163, 167, 169, 173, 179, 197, 199, 223, 227, 251, 263, 281, 289, 307, 313, 347, 359, 361, 367, 373, 379, 383, 397, 409, 419, 421, 439, 457, 461, 487, 499, 503, 523, 529, 547, 563, 577, 593

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-163))/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,163) >= 0 and the squares of primes such that (p,163) = -1, where (p,163) is the Legendre symbol.
For primes p such that (p,163) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-163))/2], namely (x + y*(1+sqrt(-163))/2) and (x + y*(1-sqrt(-163))/2), where (x,y) is a solution to x^2 + x*y + 41*y^2 = p; for p = 163, (sqrt(-163)) is the unique ideal with norm p; for primes p with (p,163) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-163))/2) = norm((1 - sqrt(-163))/2) = 41;
norm((3 + sqrt(-163))/2) = norm((3 - sqrt(-163))/2) = 43;
norm((5 + sqrt(-163))/2) = norm((5 - sqrt(-163))/2) = 47;
norm((7 + sqrt(-163))/2) = norm((7 - sqrt(-163))/2) = 53;
...
norm((79 + sqrt(-163))/2) = norm((79 - sqrt(-163))/2) = 1601.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011615, A257362, A296921, A296915.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318983.
The total number of elements with norm n is given by A318985.
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*), A341787 (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), this sequence (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=-163); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A296915 Primes that are not squares mod 163.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 59, 67, 73, 79, 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 181, 191, 193, 211, 229, 233, 239, 241, 257, 269, 271, 277, 283, 293, 311, 317, 331, 337, 349, 353, 389, 401, 431, 433, 443, 449, 463, 467, 479, 491, 509, 521, 541

Offset: 1

Ed Pegg Jr, Dec 22 2017

nonn

Inert rational primes in Q(sqrt -163). (Note that 41 is not inert in this field, it decomposes - see A296921.)

Helmut Hasse, Number Theory, Grundlehren 229, Springer, 1980, page 498.

Load the Maple program HH given in A296920. Then run HH(-163,200); - N. J. A. Sloane, Dec 26 2017

lista(nn) = forprime(p=2, nn, if (!issquare(Mod(p, 163)), print1(p, ", "));); \\ Michel Marcus, Dec 24 2017

Corrected by N. J. A. Sloane, Dec 25 2017 (including deletion of incorrect comments in CROSS-REFERENCES)

cp's OEIS Frontend

A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).

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