|
%I M5224 N2274 #40 Apr 11 2024 10:10:54
%S 31,307,643,5113,21787,39199,360007,360007,4775569,10318249,10318249,
%T 65139031,387453811,913900417,2278522747,2741702809,25147657981,
%U 118748663779,156776294593,747206701687,1151810360731,1151810360731,1151810360731
%N a(n) is the smallest prime p such that each of the first n primes has three cube roots mod p.
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%H A. E. Western and J. C. P. Miller, <a href="/A002223/a002223.pdf">Tables of Indices and Primitive Roots</a>, Royal Society Mathematical Tables, Vol. 9, Cambridge Univ. Press, 1968 [Annotated scans of selected pages]
%e 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
%p Primes:= [2]: pp:= 7:
%p for n from 1 to 12 do
%p for p from pp by 6 while
%p not(isprime(p) and andmap(t -> t &^ ((p-1)/3) mod p = 1, Primes))
%p do od:
%p A[n]:= p;
%p pp:= p;
%p Primes:= [op(Primes), nextprime(Primes[-1])];
%p od:
%p seq(A[i],i=1..12); # _Robert Israel_, Aug 02 2016
%t (* 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 *)
%Y 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).
%Y Cf. A002223, A002224.
%Y Subset of A014752. Except for a(1), subset of A014753. Except for a(1) and a(2), subset of A040044.
%K nonn,nice,more,changed
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _Don Reble_, Oct 09 2001
%E Name corrected by _Robert Israel_, Aug 02 2016
%E a(18)-a(23) from _Sergey Paramonov_, Apr 11 2024
|
