A055673 -id:A055673 - cp's OEIS frontend

A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25, 28, 31, 32, 34, 36, 41, 46, 47, 49, 50, 56, 62, 63, 64, 68, 71, 72, 73, 79, 81, 82, 89, 92, 94, 97, 98, 100, 103, 112, 113, 119, 121, 124, 126, 127, 128, 136, 137, 142, 144, 146, 151, 153, 158, 161, 162, 164, 167, 169, 175, 178

Offset: 1

N. J. A. Sloane

nonn

x^2 - 2y^2 has discriminant 8. - N. J. A. Sloane, May 30 2014
A positive number n is representable in the form x^2 - 2y^2 iff every prime p == 3 or 5 (mod 8) dividing n occurs to an even power.
Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m=2 (A035185). [amended by Georg Fischer, Sep 03 2020]
Also positive numbers of the form 2x^2 - y^2. If x^2 - 2y^2 = n, 2(x+y)^2 - (x+2y)^2 = n. - Franklin T. Adams-Watters, Nov 09 2009
Except 2, prime numbers in this sequence have the form p=8k+-1. According to the first comment, prime factors of the forms (8k+-3),(8k+-5) occur in x^2 - 2y^2 in even powers. If x^2 - 2y^2 is a prime number, those powers must be 0. Only factors 8k+-1 remain. Example: 137=8*17+1. - Jerzy R Borysowicz, Nov 04 2015
The product of any two terms of the sequence is a term too. A proof follows from the identity: (a^2-2b^2)(c^2-2d^2) = (2bd+ac)^2 - 2(ad+bc)^2. Example: 127*175 has form x^2-2y^2, with x=9335, y=6600. - Jerzy R Borysowicz, Nov 28 2015
Primitive terms (not a product of earlier terms that are greater than 1 in the sequence) are A055673 except 1. - Charles R Greathouse IV, Sep 10 2016
Positive numbers of the form u^2 + 2uv - v^2. - Thomas Ordowski, Feb 17 2017
For integer numbers z, a, k and z^2+a^2>0, k>=0: z^(4k) + a^4 is in A035251 because z^(4k) + a^4 = (z^(2k) + a^2)^2 - 2(a*z^k)^2. Assume 0^0 = 1. Examples: 3^4 + 1^4 = 82, 3^8+4^4=6817. - Jerzy R Borysowicz, Mar 09 2017
Numbers that are the difference between two legs of a Pythagorean right triangle. - Michael Somos, Apr 02 2017

			The (x,y) pairs, with minimum x, that solve the equation are (1,0), (2,1), (2,0), (3,1), (4,2), (3,0), (4,1), (4,0), (5,2), (6,3), (5,1), (5,0), (6,2), (7,3), (8,4), (6,1), (6,0), (7,2), (8,3), (7,1), (7,0), (10,5), (8,2), ... If the positive number is a perfect square, y=0 yields a trivial solution. - _R. J. Mathar_, Sep 10 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Matthews, Thue's theorem and the diophantine equation x^2-D*y^2=+-N, Math. Comp. 71 (239) (2002) 1281-1286.
K. Matthews, Solving the diophantine equation x^2-D*y^2=N, D>0, (2016).
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 [Edited and cached copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A035185, A042965, A001481, A000047.
Primes: A038873.
Complement of A232531. - Thomas Ordowski and Altug Alkan, Feb 09 2017

filter:= proc(n) local F;
  F:= select(t -> t[1] mod 8 = 3 or t[1] mod 8 = 5, ifactors(n)[2]);
  map(t -> t[2],F)::list(even);
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 01 2015

Reap[For[n = 1, n < 200, n++, r = Reduce[x^2 - 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

select(x -> x, direuler(p=2,201,1/(1-(kronecker(2,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 17 2006 */

is(n)=#bnfisintnorm(bnfinit(z^2-2),n) \\ Ralf Stephan, Oct 14 2013

from itertools import count, islice
from sympy import factorint
def A035251_gen(): # generator of terms
    return filter(lambda n:all(not((2 < p & 7 < 7) and e & 1) for p, e in factorint(n).items()),count(1))
A035251_list = list(islice(A035251_gen(),30)) # Chai Wah Wu, Jun 28 2022

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

A091727 Norms of prime ideals of Z[sqrt(-5)].

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 121, 127, 149, 163, 167, 169, 181, 223, 227, 229, 241, 263, 269, 281, 283, 289, 307, 347, 349, 361, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487

Offset: 1

Paul Boddington, Feb 02 2004

Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.
From Jianing Song, Feb 20 2021: (Start)
The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[sqrt(-5)] has class number 2.
For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.
For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.
For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

			From _Jianing Song_, Feb 20 2021: (Start)
Let |I| be the norm of an ideal I, then:
|(2, 1+sqrt(-5))| = 2;
|(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;
|(sqrt(-5))| = 5;
|(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;
|(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;
|(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;
|(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)

David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.
A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.

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

Cf. A091728.
Cf. A289741, A033205, A106865, A139513, A003626.
The number of distinct ideals with norm n is given by A035170.
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), this sequence (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.

isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n,&p)); if(!e || e>2, 0, bitxor(e-1,!!vecsearch(ms,p%20))); }; \\ Antti Karttunen, Feb 24 2020

Offset corrected by Jianing Song, Feb 20 2021

A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

Original entry on oeis.org

2, 3, 11, 17, 19, 25, 41, 43, 49, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 169, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 529, 547, 563

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[sqrt(-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 congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.
For primes p == 1, 3 (mod 8), there are two distinct ideals with norm p in Z[sqrt(2)], namely (x + y*sqrt(-2)) and (x - y*sqrt(-2)), where (x,y) is a solution to x^2 + 2*y^2 = p; for p = 2, (sqrt(-2)) is the unique ideal with norm p; for p == 5, 7 (mod 8), (p) is the only ideal with norm p^2.

			norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;
norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;
norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;
norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.

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

Cf. A188510, A033203, A033200, A003628.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.
The total number of elements with norm n is given by A033715.
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), this sequence (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (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.

isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

Original entry on oeis.org

3, 4, 5, 11, 23, 31, 37, 47, 49, 53, 59, 67, 71, 89, 97, 103, 113, 137, 157, 163, 169, 179, 181, 191, 199, 223, 229, 251, 257, 269, 289, 311, 313, 317, 331, 353, 361, 367, 379, 383, 389, 397, 401, 419, 421, 433, 443, 449, 463, 467, 487, 499, 509, 521

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-11))/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 congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.
For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.

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

Cf. A011582, A056874, A296920, A191060.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.
The total number of elements with norm n is given by A028609.
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), this sequence (D=-11), A341786 (D=-15*), A341787 (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.

isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).

Original entry on oeis.org

2, 3, 5, 17, 19, 23, 31, 47, 49, 53, 61, 79, 83, 107, 109, 113, 121, 137, 139, 151, 167, 169, 173, 181, 197, 199, 211, 227, 229, 233, 241, 257, 263, 271, 293, 317, 331, 347, 349, 353, 379, 383, 409, 421, 439, 443, 467, 499, 503, 541, 557, 563, 571, 587

Offset: 1

Jianing Song, Feb 19 2021

The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Note that Z[(1+sqrt(-15))/2] has class number 2.
Consists of the primes congruent to 1, 2, 3, 4, 5, 8 modulo 15 and the squares of primes congruent to 7, 11, 13, 14 modulo 15.
For primes p == 1, 4 (mod 15), there are two distinct ideals with norm p in Z[(1+sqrt(-15))/2], namely (x + y*(1+sqrt(-15))/2) and (x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p; for p == 2, 8 (mod 15), there are also two distinct ideals with norm p, namely (p, x + y*(1+sqrt(-15))/2) and (p, x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p^2 with y != 0; (3, sqrt(-15)) and (5, sqrt(-15)) are respectively the unique ideal with norm 3 and 5; for p == 7, 11, 13, 14 (mod 15), (p) is the only ideal with norm p^2.

			Let |I| be the norm of an ideal I, then:
|(2, (1+sqrt(-15))/2)| = |(2, (1-sqrt(-15))/2)| = 2;
|(3, sqrt(-15))| = 3;
|(5, sqrt(-15))| = 5;
|(17, 7+4*sqrt(-15))| = |(17, 7-4*sqrt(-15))| = 17;
|(2 + sqrt(-15))| = |(2 - sqrt(-15))| = 19;
|(23, 17+4*sqrt(-15))| = |(23, 17-4*sqrt(-15))| = 23;
|(4 + sqrt(-15))| = |(4 - sqrt(-15))| = 31.

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

Cf. A316569, A033212, A106859, A191018, A191062.
The number of distinct ideals with norm n is given by A035175.
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), this sequence (D=-15*), A341787 (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.

isA341786(n) = my(disc=-15); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

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

Original entry on oeis.org

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

Offset: 1

Jianing Song, Feb 19 2021

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.

			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.

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

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.

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

A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

Original entry on oeis.org

4, 9, 11, 13, 17, 23, 25, 31, 41, 43, 47, 49, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139, 167, 173, 181, 193, 197, 229, 239, 251, 269, 271, 281, 283, 293, 307, 311, 317, 337, 353, 359, 361, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-43))/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,43) >= 0 and the squares of primes such that (p,43) = -1, where (p,43) is the Legendre symbol.
For primes p such that (p,43) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-43))/2], namely (x + y*(1+sqrt(-43))/2) and (x + y*(1-sqrt(-43))/2), where (x,y) is a solution to x^2 + x*y + 11*y^2 = p; for p = 43, (sqrt(-43)) is the unique ideal with norm p; for primes p with (p,43) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-43))/2) = norm((1 - sqrt(-43))/2) = 11;
norm((3 + sqrt(-43))/2) = norm((3 - sqrt(-43))/2) = 13;
norm((5 + sqrt(-43))/2) = norm((5 - sqrt(-43))/2) = 17;
norm((7 + sqrt(-43))/2) = norm((7 - sqrt(-43))/2) = 23;
...
norm((19 + sqrt(-43))/2) = norm((19 - sqrt(-43))/2) = 101.

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

Cf. A011591, A106891, A191031, A184902.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035147.
The total number of elements with norm n is given by A138811.
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*), this sequence (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341788(n) = my(disc=-43); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A341789 Norms of prime elements in Z[(1+sqrt(-67))/2], the ring of integers of Q(sqrt(-67)).

Original entry on oeis.org

4, 9, 17, 19, 23, 25, 29, 37, 47, 49, 59, 67, 71, 73, 83, 89, 103, 107, 121, 127, 131, 149, 151, 157, 163, 167, 169, 173, 181, 193, 199, 211, 223, 227, 241, 257, 263, 269, 277, 283, 293, 307, 317, 349, 359, 389, 397, 419, 421, 431, 439, 449, 457, 461

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-67))/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,67) >= 0 and the squares of primes such that (p,67) = -1, where (p,67) is the Legendre symbol.
For primes p such that (p,67) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-67))/2], namely (x + y*(1+sqrt(-67))/2) and (x + y*(1-sqrt(-67))/2), where (x,y) is a solution to x^2 + x*y + 17*y^2 = p; for p = 67, (sqrt(-67)) is the unique ideal with norm p; for primes p with (p,67) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-67))/2) = norm((1 - sqrt(-67))/2) = 17;
norm((3 + sqrt(-67))/2) = norm((3 - sqrt(-67))/2) = 19;
norm((5 + sqrt(-67))/2) = norm((5 - sqrt(-67))/2) = 23;
norm((7 + sqrt(-67))/2) = norm((7 - sqrt(-67))/2) = 29;
...
norm((31 + sqrt(-67))/2) = norm((31 - sqrt(-67))/2) = 257.

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

Cf. A011596, A106933, A191041, A191077.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318982.
The total number of elements with norm n is given by A318984.
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), this sequence (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=-67); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

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

Original entry on oeis.org

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)

A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

Original entry on oeis.org

4, 5, 9, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 169, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 289, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 529

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(5))/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 congruent to 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.
For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

			norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;
norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;
norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;
norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

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

Cf. A080891, A038872, A045468, A003631.
The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.
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), this sequence (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), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

cp's OEIS Frontend

A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

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A091727 Norms of prime ideals of Z[sqrt(-5)].

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A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

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A341785 Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).

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A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).

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

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A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

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A341789 Norms of prime elements in Z[(1+sqrt(-67))/2], the ring of integers of Q(sqrt(-67)).