A147972 -id:A147972 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

			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

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.

A. E. Western and J. C. P. Miller, Tables of Indices and Primitive Roots, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]

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.

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

(* 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 *)

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

A191089 Least prime p such that the first n primes are not squares mod p.

Original entry on oeis.org

3, 5, 43, 43, 67, 67, 163, 163, 163, 163, 163, 163, 74093, 170957, 360293, 679733, 2004917, 2004917, 51599563, 51599563, 96295483, 96295483, 146161723, 1408126003, 1728061733, 1728061733, 1728061733, 1728061733, 1728061733, 1728061733

Offset: 1

T. D. Noe, May 25 2011

nonn

That is, the first n primes are quadratic non-residues mod p. A less restrictive form of A001992. Sequence A191088 is similar, but allows p to be composite. See A147972 for the square version.

Cf. A001992, A147972, A191088.

Table[p = 2; While[Length[Select[Prime[Range[n]], JacobiSymbol[#, p] == -1 &]] < n, p = NextPrime[p]]; p, {n, 15}]

q=2;forprime(k=3,1e9,forprime(p=2,q,if(kronecker(p,k)>=0,next(2)));print1(k", ");q=nextprime(q+1);k--) \\ Charles R Greathouse IV, Oct 10 2011

a(16)-a(30) from Charles R Greathouse IV, Oct 10 2011

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions