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.

A002225 a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.

Original entry on oeis.org

31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

a(n) is the smallest prime p == 1 (mod 3) such that each of the first n primes is a cubic residue mod p. - Robert Israel, Aug 02 2016

Examples

			For n = 2, the first two primes 2 and 3 each have three cube roots mod 307: 2 has cube roots 52, 270, 292 and 3 has cube roots 192, 194, 228. - _Robert Israel_, Aug 02 2016

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. XVI.

Links

Crossrefs

Smallest prime p such that each of the first n primes has q q-th roots mod p: A147972 (q=2), this sequence (q=3), A002226 (q=5), A002227 (q=7), A002228 (q=11), A060363 (q=13), A060364 (q=17).
Cf. A002223, A002224.
Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.

Programs

  • Maple
    Primes:= [2]: pp:= 7:
for n from 1 to 12 do
  for p from pp by 6 while
    not(isprime(p) and andmap(t -> t &^ ((p-1)/3) mod p = 1, Primes))
  do od:
  A[n]:= p;
  pp:= p;
  Primes:= [op(Primes), nextprime(Primes[-1])];
od:
seq(A[i],i=1..12); # Robert Israel, Aug 02 2016
  • Mathematica
    (* This naive program being very slow, limit is set to 8 terms *) lim=8; np[] := While[p=NextPrime[p]; Mod[p,3]!=1]; crQ[n_, p_] := Reduce[ 0A002225={}; While[Length[A002225] < lim, If[And @@ (crQ[#,p]& /@ pp), AppendTo[pp, NextPrime[ Last[pp]]]; Print[p]; AppendTo[A002225,p], np[] ] ]; A002225 (* Jean-François Alcover, Sep 09 2011 *)

Extensions

More terms from Don Reble, Oct 09 2001
Name corrected by Robert Israel, Aug 02 2016
a(18)-a(23) from Sergey Paramonov, Apr 11 2024

A002228 Smallest prime p such that first n primes (p_1=2, ..., p_n) are 11th power residues mod p.

Original entry on oeis.org

331, 39139, 253243, 4397207, 21587171, 781712537, 781712537, 25467966877, 1304374210679, 4331892405391
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. XXIV.

Links

Crossrefs

Cf. A002223-A002227, A060363, A060364.

Extensions

More terms from Don Reble, Oct 12 2001
a(9)-a(10) from Sergey Paramonov, Apr 14 2024

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

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 1-4 of 4 results.

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

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