cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A296926 Rational primes that decompose in the field Q(sqrt(-13)).

Original entry on oeis.org

7, 11, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631, 641, 643, 653, 673, 677, 683, 691
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

In general, primes that decompose in Q(sqrt(-p prime)) are congruent modulo 4p to t(-1)^[t^(phi(p)/2) mod p = 1 XOR t mod min(e,4) = 1], where t are the totatives of 2p, e is the even part of phi(p), and [P] returns 1 if P else 0. In other words, if phi(p) is at least twice even, then the t are signed so that the quadratic residuosity of t mod p aligns with the congruence of +-t mod 4 to 1--the modulus 4p is thence irreducible--; if only once, then the signature simply indicates quadratic residues modulo p. The imbalance of signs in either flank (t < p, t > p) of the signature also gives the class number of Q(sqrt(-p)), up to an excess factor of 3 if p == 3 (mod 8) but != 3. [E.g., for p = 13 we have +--+++ or +++--+, so the class number of Q(sqrt(-13)) = 2; for p = 11 == 3 (mod 8) we have +++-+ or -+---, so the class number of Q(sqrt(-11)) = 3/3 = 1.] - Travis Scott, Jan 05 2023

Links

Crossrefs

Cf. A105885, A296920, A296927, A296928.

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.
  • Mathematica
    Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* Amiram Eldar, Nov 17 2023 *)
  • PARI
    list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(-13,p)==1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 18 2018

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018
Primes == {1, 7, 9, 11, 15, 17, 19, 25, 29, 31, 47, 49} (mod 52). - Travis Scott, Jan 05 2023

A296927 Inert rational primes in the field Q(sqrt(-13)).

Original entry on oeis.org

3, 5, 23, 37, 41, 43, 73, 79, 89, 97, 103, 107, 109, 127, 131, 137, 139, 149, 179, 191, 193, 197, 199, 211, 229, 241, 251, 263, 281, 283, 293, 311, 317, 347, 349, 353, 367, 397, 401, 409, 419, 421, 439, 443, 449, 457, 461, 467, 491, 503, 509, 523, 541, 547, 557, 563
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Links

Programs

A296928 Primes p such that Legendre(-13,p) = 0 or 1.

Original entry on oeis.org

7, 11, 13, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631
Offset: 1

Views

Author

N. J. A. Sloane, Dec 26 2017

Keywords

Comments

Primes == 1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, or 49 (mod 52). - Robert Israel, Dec 27 2017

Links

Programs

  • Maple
    Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.
select(isprime, [seq(seq(52*i+j, j=[1, 7, 9, 11, 13, 15, 17, 19, 25, 29, 31, 47, 49]),i=0..50)]); # Robert Israel, Dec 27 2017

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587
Offset: 1

Views

Author

Dimitris Cardaris, Feb 03 2024

Keywords

Comments

Primes p such that Legendre(-21,p) = -1.

Crossrefs

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Programs

  • Mathematica
    Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)
  • SageMath
    [p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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