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A002225
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a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.
(Formerly M5224 N2274)
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6
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31, 307, 643, 5113, 21787, 39199, 360007, 360007, 4775569, 10318249, 10318249, 65139031, 387453811, 913900417, 2278522747, 2741702809, 25147657981, 118748663779, 156776294593, 747206701687, 1151810360731, 1151810360731, 1151810360731
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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
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MAPLE
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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:
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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KEYWORD
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nonn,nice,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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