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.

A003628 Primes congruent to {5, 7} mod 8.

Original entry on oeis.org

5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 263, 269, 271, 277, 293, 311, 317, 349, 359, 367, 373, 383, 389, 397, 421, 431, 439
Offset: 1

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

Inert rational odd primes in the field Q(sqrt(-2)).
Primes p such that p XOR 5 = p - 5. - Brad Clardy, Jul 22 2012
Terms m in A047566 with A010051(m) = 1. - Reinhard Zumkeller, Dec 29 2012
This sequence gives the primes p which satisfy norm(rho(p)) = - 1 with rho(p) := 2*cos(Pi/p) (the length ratio (smallest diagonal)/side in the regular p-gon). The norm of an algebraic number (over Q) is the product over all zeros of its minimal polynomial. Here norm(rho(p)) = (-1)^delta(p)* C(p, 0), with the degree delta(p) = A055034(p) = (p-1)/2. For p == 5 (mod 8) the norm is C(p, 0) (see a comment on 2*A230076) and for p == 7 (mod 8) the norm is -C(p, 0) (see a comment on A186302). For the primes with norm(rho(p)) = +1 see A033200. - Wolfdieter Lang, Oct 24 2013

References

  • H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000040, A039706, A033203 (complement with respect to A000040).

Programs

  • Haskell
    a003628 n = a003628_list !! (n-1)
a003628_list = filter ((== 1) . a010051) a047566_list
-- Reinhard Zumkeller, Dec 29 2012, Jan 22 2012
  • Magma
    [ p: p in PrimesUpTo(600) | p mod 8 in {5, 7}]; // Vincenzo Librandi, Aug 22 2012
  • Mathematica
    Select[Prime[Range[200]],MemberQ[{5,7},Mod[#,8]]&] (* Harvey P. Dale, Oct 24 2011 *)
  • PARI
    {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( -2, m )==-1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 24 2023

A381256 Numbers k such that 5*k+1 divides 5^k+1.

Original entry on oeis.org

0, 1, 625, 57057, 7748433, 30850281, 111494625, 393423745, 499088601, 519341361, 1051107705, 1329416385, 1616038425, 2215448001, 2433936225, 2852972265, 3399207273, 4344683849, 4961725281, 5454760185, 5485530369, 6578054145, 6678031745, 7701979761, 7807302825
Offset: 1

Views

Author

René-Louis Clerc, Feb 18 2025

Keywords

Comments

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

Examples

			5*625+1 = 3126 divides 5^625+1.

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Links

Crossrefs

Cf. A224486, A222948, A230076, A381257, A381258.

Programs

  • PARI
    isok(n) = my(m=5*n+1); Mod(5, m)^n==-1

A381257 Numbers k such that 6*k+1 divides 6^k+1.

Original entry on oeis.org

0, 1, 6, 30, 58, 70, 73, 90, 101, 105, 121, 125, 146, 153, 166, 170, 181, 182, 185, 210, 233, 241, 242, 266, 282, 290, 322, 373, 381, 385, 390, 397, 441, 445, 446, 450, 453, 530, 557, 562, 585, 593, 601, 602, 605, 606, 621, 646, 653, 670, 685, 710, 726, 805, 810, 817, 833, 837, 853, 866
Offset: 1

Views

Author

René-Louis Clerc, Feb 18 2025

Keywords

Comments

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

Examples

			6*30+1 = 181 divides 6^30+1 = 221073919720733357899777.

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Links

Crossrefs

Cf. A224486, A222948, A230076, A381256, A381258.

Programs

  • Mathematica
    Select[Range[0, 866], PowerMod[6, #, 6#+1]==6#&]  (* James C. McMahon, Apr 02 2025 *)
  • PARI
    isok(n) = my(m=6*n+1); Mod(6, m)^n==-1

A381258 Numbers k such that 7*k+1 divides 7^k+1.

Original entry on oeis.org

0, 1, 135, 5733, 11229, 42705, 50445, 117649, 131365, 168093, 636405, 699825, 1269495, 2528155, 4226175, 6176709, 6502545, 9365265, 9551115, 13227021, 14464485, 14912625, 20859435, 26903605, 28251265, 30589905, 32660901, 37597329, 41506875, 42766465, 55452075, 56192535, 111898605
Offset: 1

Views

Author

René-Louis Clerc, Feb 18 2025

Keywords

Comments

The numbers are called Curzon numbers by Tattersall (p. 85, exercise 43).

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005, p. 85.

Links

Crossrefs

Cf. A224486, A222948, A230076, A381256, A381257.

Programs

  • Mathematica
    Select[Range[0,10^7],PowerMod[7,#,7#+1]==7#&] (* James C. McMahon, Mar 05 2025 *)
  • PARI
    isok(n) = my(m=7*n+1); Mod(7, m)^n==-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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