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 1-3 of 3 results.

A191065 Primes that are not squares mod 23.

Original entry on oeis.org

5, 7, 11, 17, 19, 37, 43, 53, 61, 67, 79, 83, 89, 97, 103, 107, 109, 113, 137, 149, 157, 181, 191, 199, 227, 229, 241, 251, 263, 281, 283, 293, 313, 337, 359, 367, 373, 379, 383, 389, 401, 419, 421, 431, 433, 457, 467, 479, 503, 521, 523, 557, 563, 569, 571
Offset: 1

Views

Author

T. D. Noe, May 25 2011

Keywords

Comments

Inert rational primes in the field Q(sqrt(-23)). - N. J. A. Sloane, Dec 26 2017
It appears that this is the same as primes p such that A000594(p) = RamanujanTau(p) == 0 (mod 23). - Ray Chandler, Dec 01 2016

Links

Crossrefs

Primes in A278579.

Programs

  • Magma
    [p: p in PrimesUpTo(571) | JacobiSymbol(p, 23) eq -1]; // Vincenzo Librandi, Sep 11 2012
  • Mathematica
    Select[Prime[Range[200]], JacobiSymbol[#,23]==-1&]

A278580 Numbers n such that Jacobi(n,23) = 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41, 47, 48, 49, 50, 52, 54, 55, 58, 59, 62, 64, 70, 71, 72, 73, 75, 77, 78, 81, 82, 85, 87, 93, 94, 95, 96, 98, 100, 101, 104, 105, 108, 110, 116, 117, 118, 119, 121, 123, 124, 127, 128, 131, 133, 139, 140, 141, 142, 144, 146
Offset: 1

Views

Author

N. J. A. Sloane, Nov 29 2016

Keywords

Comments

Important for the study of Ramanujan numbers A000594.
The first 11 terms, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, are the quadratic residues mod 23 (see row 23 of A063987).

Links

Crossrefs

Cf. A010385, A000594, A063987, A278579.

Programs

  • Mathematica
    LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1},{1,2,3,4,6,8,9,12,13,16,18,24},90] (* Harvey P. Dale, Jun 25 2020 *)
  • PARI
    Vec(x*(1+x+x^2+x^3+2*x^4+2*x^5+x^6+3*x^7+x^8+3*x^9+2*x^10+5*x^11) / ((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10)) + O(x^100)) \\ Colin Barker, Nov 30 2016

Formula

From Colin Barker, Nov 30 2016: (Start)
a(n+11) = a(n) + 23.
a(n) = a(n-1) + a(n-11) - a(n-12) for n>12.
G.f.: x*(1 +x +x^2 +x^3 +2*x^4 +2*x^5 +x^6 +3*x^7 +x^8 +3*x^9 +2*x^10 +5*x^11) / ((1 -x)^2*(1 +x +x^2 +x^3 +x^4 +x^5 +x^6 +x^7 +x^8 +x^9 +x^10))
(End)

A028736 Nonsquares mod 23.

Original entry on oeis.org

5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Since 11 is not a perfect square and there are no solutions to x^2 = 11 mod 23, 11 is in the sequence.
Although 12 is not a perfect square either, there are solutions to x^2 = 12 mod 23, such as x = 9, x = 14. Thus 12 is not in the sequence.

References

  • Srinivasa Ramanujan, Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962. See "Congruence properties of partitions", p. 230. - N. J. A. Sloane, Jun 01 2014

Links

Crossrefs

Cf. A010385, A028759, A278579.

Programs

  • Mathematica
    Select[Range[22], JacobiSymbol[#, 23] != 1 &] (* Jean-François Alcover, Oct 07 2018 *)
  • Scala
    val squaresMod23 = (0 to 22).map(n => n * n).map(_ % 23)
(0 to 22).diff(squaresMod23) // Alonso del Arte, Nov 23 2019
Showing 1-3 of 3 results.

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

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