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 1-5 of 5 results.

A106867 Primes of the form 2*x^2 + x*y + 3*y^2.

Original entry on oeis.org

2, 3, 13, 29, 31, 41, 47, 71, 73, 127, 131, 139, 151, 163, 179, 193, 197, 233, 239, 257, 269, 277, 311, 331, 349, 353, 397, 409, 439, 443, 461, 487, 491, 499, 509, 541, 547, 577, 587, 601, 647, 653, 673, 683, 739, 761, 811, 823, 857, 859, 863, 887, 929, 947
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant = -23.
Primes p such that the polynomial x^3-x-1 is irreducible over Zp. The polynomial discriminant is also -23. - T. D. Noe, May 13 2005
Also, primes p such that tau(p) = A000594(p) == -1 (mod 23). [A proof can probably be found in van der Blij (1952). Thanks to Juan Arias-de-Reyna for this reference. - N. J. A. Sloane, Nov 29 2016]

References

  • F. van der Blij, Binary quadratic forms of discriminant -23. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math. 14, (1952). 498-503; Math. Rev. MR0052462.
  • John Raymond Wilton, "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. The primes are listed in Table II.

Links

Crossrefs

Cf. A086965 (number of distinct zeros of x^3-x-1 mod prime(n)).
Cf. also A000594.
These are the primes in A028929.

Programs

  • Mathematica
    Union[QuadPrimes2[2, 1, 3, 10000], QuadPrimes2[2, -1, 3, 10000]] (* see A106856 *)
  • PARI
    forprime(p=2,10^4,if(0==#polrootsmod(x^3-x-1,p),print1(p,", "))); /* Joerg Arndt, Jul 27 2011 */
  • PARI
    forprime(p=2,10^4,if(polisirreducible(Mod(1, p)*(x^3-x-1)), print1(p, ", ") ) ); /* Joerg Arndt, Mar 30 2013 */
  • Python
    from itertools import count, islice
from sympy import prime, GF, Poly
from sympy.abc import x
def A106867_gen(): # generator of terms
    return filter(lambda p:Poly(x**3-x-1,domain=GF(p)).is_irreducible, (prime(i) for i in count(1)))
A106867_list = list(islice(A106867_gen(),20)) # Chai Wah Wu, Nov 11 2022

A028951 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 4 ] (or the Kleinian 2-d lattice, see A002652).

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 9, 11, 14, 16, 18, 22, 23, 25, 28, 29, 32, 36, 37, 43, 44, 46, 49, 50, 53, 56, 58, 63, 64, 67, 71, 72, 74, 77, 79, 81, 86, 88, 92, 98, 99, 100, 106, 107, 109, 112, 113, 116, 121, 126, 127, 128, 134, 137, 142, 144, 148, 149, 151, 154, 158, 161, 162
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Or, numbers of the form x^2+xy+2y^2 with x and y integers. - N. J. A. Sloane, Apr 30 2015

Links

Crossrefs

Cf. A028929, A035248, A002652, A034036, A257346 (complement).

Programs

  • Mathematica
    Reap[For[n = 0, n < 200, n++, r = Reduce[x^2 + x y + 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A028958 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 12 ] (divided by 2).

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 16, 18, 23, 24, 25, 26, 27, 32, 36, 39, 48, 49, 52, 54, 58, 59, 62, 64, 72, 78, 81, 82, 87, 92, 93, 94, 96, 100, 101, 104, 108, 116, 117, 121, 123, 124, 128, 138, 141, 142, 144, 146, 150, 156, 162, 164, 167, 169, 173, 174, 184, 186, 188, 192
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Nonnegative integers of the form x^2 + x*y + 6*y^2, discriminant -23. - Ray Chandler, Jul 12 2014
The Gram matrix is positive-definite, therefore, if w := (1 + sqrt(-23)) / 2, then |x + w*y|^2 = x^2 + x*y + 6*y^2 > 0 for all integers x and y except x = y = 0. - Michael Somos, Mar 28 2015
The theta function of the lattice with basis [1, w] is the g.f. of A028959, therefore, A028959(n) is positive if and only if n is in this sequence. - Michael Somos, Mar 28 2015

Links

Crossrefs

For primes see A033217. Cf. A028929, A106867.
Cf. A028959.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A106869 Primes of the form x^2+xy+6y^2, with x and y nonnegative.

Original entry on oeis.org

59, 101, 167, 173, 223, 271, 307, 317, 347, 449, 463, 593, 599, 607, 691, 719, 809, 821, 877, 883, 991, 997, 1097, 1117, 1151, 1181, 1231, 1319, 1451, 1453, 1553, 1613, 1669, 1697, 1787, 1789, 1871, 1889, 1913, 2027, 2053, 2143, 2339, 2347
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant=-23.

Links

Crossrefs

Cf. A028929, A106867.

Programs

  • Mathematica
    QuadPrimes2[1, 1, 6, 10000] (* see A106856 *)

A106868 Primes of the form 2x^2-xy+3y^2, with x and y nonnegative.

Original entry on oeis.org

2, 3, 29, 31, 47, 73, 131, 151, 163, 193, 197, 233, 239, 277, 349, 353, 397, 487, 491, 499, 509, 547, 577, 601, 647, 653, 683, 811, 857, 859, 863, 929, 947, 1013, 1021, 1039, 1093, 1283, 1289, 1291, 1297, 1301, 1327, 1361, 1429, 1499, 1511, 1531
Offset: 1

Views

Author

T. D. Noe, May 09 2005

Keywords

Comments

Discriminant=-23.

Links

Crossrefs

Cf. A028929, A106867.

Programs

  • Mathematica
    QuadPrimes2[2, -1, 3, 10000] (* see A106856 *)
Showing 1-5 of 5 results.

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

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