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A002225
a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.
(Formerly M5224 N2274)
6
31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731
OFFSET
1,1
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
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
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]
EXAMPLE
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
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[ 0<x<p && Mod[x^3, p]==n, x, Integers]=!=False; p=2; pp={p}; A002225={}; 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 *)
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).
Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.
Sequence in context: A141912 A147297 A295975 * A124335 A368953 A368946
KEYWORD
nonn,nice,more
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
STATUS
approved