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.

A002223 Smallest prime p of form p = 8k-1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p.

Original entry on oeis.org

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 4080359, 12537719, 30706079, 36415991, 82636319, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 33857579279
Offset: 1

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Author

N. J. A. Sloane

Keywords

Examples

			12^2 = 2 mod 71, 28^2 = 3 mod 71, 17^2 = 5 mod 71.

References

  • N. D. Bronson and D. A. Buell, Congruential sieves on FPGA computers, pp. 547-551 of Mathematics of Computation 1943-1993 (Vancouver, 1993), Proc. Symp. Appl. Math., Vol. 48, Amer. Math. Soc. 1994.
  • D. H. Lehmer, E. Lehmer and D. Shanks, Integer sequences having prescribed quadratic character, Math. Comp., 24 (1970), 433-451.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XV.

Links

Crossrefs

Cf. A045535, A002224, A002225.

Programs

  • Mathematica
    np[] := While[p = NextPrime[p]; Mod[p, 8] != 7]; p = 2; A002223 = {}; pp = {2}; np[]; While[ Length[A002223] < 26, If[Union[ JacobiSymbol[#, p] &[pp]] === {1}, AppendTo[pp, NextPrime[Last[pp]]]; Print[p]; AppendTo[A002223, p], np[]]]; A002223 (* Jean-François Alcover, Sep 09 2011 *)

Extensions

The Bronson-Buell reference gives terms through 227.
More terms from Don Reble, Sep 19 2001

A002224 Smallest prime p of form p = 8k+1 such that first n primes (p_1=2, ..., p_n) are quadratic residues mod p.

Original entry on oeis.org

17, 73, 241, 1009, 2689, 8089, 33049, 53881, 87481, 483289, 515761, 1083289, 3818929, 3818929, 9257329, 22000801, 48473881, 48473881, 175244281, 427733329, 427733329, 898716289, 8114538721, 9176747449, 23616331489, 23616331489, 23616331489, 196265095009, 196265095009, 196265095009, 196265095009, 2871842842801, 2871842842801, 2871842842801, 26437680473689
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			32^2 = 2 mod 73, 21^2 = 3 mod 73.

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XV.

Links

Crossrefs

Cf. A002223, A002225, A002226.

Programs

  • Mathematica
    f[n_] := Block[{k = 2}, While[JacobiSymbol[n, Prime[k]] == 1, k++ ]; Prime[k]] (* Robert G. Wilson v *)
np[] := While[p = NextPrime[p]; Mod[p, 8] != 1]; p = 2; A002224 = {}; pp = {2}; np[]; While[Length[A002224] < 25, If[Union[JacobiSymbol[#, p] &[pp]] === {1}, AppendTo[pp, NextPrime[Last[pp]]]; Print[p]; AppendTo[A002224, p], np[]]]; A002224 (* Jean-François Alcover, Sep 09 2011 *)
  • PARI
    a(n,startAt=17)=my(v=primes(n)); forprime(p=startAt,, if(p%8>1, next); for(i=1,n, if(kronecker(v[i],p)<1, next(2))); return(p)) \\ Charles R Greathouse IV, Jun 26 2017

Extensions

More terms from Don Reble, Sep 19 2001
More terms from Mike Oakes, Nov 28 2022

A002226 Smallest prime p such that first n primes (p_1=2, ..., p_n) are quintic residues mod p.

Original entry on oeis.org

151, 431, 6581, 67651, 241981, 2081921, 3395921, 116900011, 650086271, 858613901, 11736494711, 50888057851, 303855349271, 2459339487751, 3167880361091
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XXIII.

Links

Crossrefs

Cf. A002223, A002224, A002225, A002227, etc.

Extensions

More terms from Don Reble, Oct 10 2001
a(13)-a(15) from Sergey Paramonov, Apr 08 2024

A002227 Smallest prime p such that first n primes (p_1=2, ..., p_n) are 7th power residues mod p.

Original entry on oeis.org

631, 5531, 72661, 865957, 2375059, 32353609, 175175603, 945552637, 945552637, 54144188771, 688203780167, 2701344818803
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(13) > 2^42. - Sergey Paramonov, Apr 12 2024

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots. Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968, p. XXIII.

Links

Crossrefs

Cf. A002223, A002224, A002225, A002226, A002228.

Extensions

More terms from Don Reble, Oct 12 2001
a(11)-a(12) from Sergey Paramonov, Apr 12 2024

A147972 Smallest prime p modulo which the first n primes are nonzero quadratic residues.

Original entry on oeis.org

7, 23, 71, 311, 479, 1559, 5711, 10559, 18191, 31391, 366791, 366791, 366791, 3818929, 9257329, 22000801, 36415991, 48473881, 120293879, 120293879, 131486759, 131486759, 2929911599, 2929911599, 7979490791, 23616331489, 23616331489, 89206899239, 121560956039, 196265095009, 196265095009, 513928659191, 5528920734431, 8402847753431, 8402847753431, 8402847753431, 70864718555231
Offset: 1

Views

Author

Max Alekseyev, Nov 18 2008

Keywords

Comments

The same primes without repetitions are listed in A147970.
a(n) <= min{A002223(n), A002224(n)}. What is the smallest n for which this inequality is strict?
By definition, a(n) == 1, 7 (mod 8), so a(n) = min{A002223(n), A002224(n)}. - Jianing Song, Feb 18 2019

Crossrefs

Cf. A000229, A002189, A002223, A002224, A045535, A053760, A133435, A147969, A147970, A147971.
Smallest prime p such that each of the first n primes has q q-th roots mod p: this sequence (q=2), A002225 (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).

Programs

  • Mathematica
    (*version 7.0*)m=1;P=7;Lst={p};While[m<25,m++;S=Prime[Range[m]];While[MemberQ[JacobiSymbol[S,p],-1],p=NextPrime[p]];Lst=Append[Lst,P]];Lst (* Emmanuel Vantieghem, Jan 31 2012 *)
  • PARI
    t=2;forprime(p=2,1e9,forprime(q=2,t,if(kronecker(q,p)<1,next(2)));print1(p", ");t=nextprime(t+1);p--) \\ Charles R Greathouse IV, Jan 31 2012

Formula

a(n) >= min{A002189(n-1), A045535(n-1)}. - Jianing Song, Feb 18 2019

Extensions

a(23)-a(25) from Emmanuel Vantieghem, Jan 31 2012
a(26)-a(37) from Max Alekseyev, Aug 21 2015
Showing 1-5 of 5 results.

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

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