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A205989 a(n) = smallest prime >= 10^n with primitive root 10. 1


%S 7,17,109,1019,10007,100019,1000171,10000019,100000007,1000000007,

%T 10000000019,100000000019,1000000000061,10000000000051,

%U 100000000000097,1000000000000091,10000000000000061,100000000000000019,1000000000000000177,10000000000000000051

%N a(n) = smallest prime >= 10^n with primitive root 10.

%C From _David W. Wilson_, Feb 17 2012 : (Start)

%C The decimal expansion of 1/a(n) includes every possible block of n digits. Conjecturally, a(n) is the smallest value with this property.

%C If Artin's conjecture is true, there are an infinite number of primes with primitive root 10, which implies that a(n) exists for all n. Artin's conjecture remains open. (End)

%H David W. Wilson, <a href="/A205989/b205989.txt">Table of n, a(n) for n = 0..75</a>

%H <a href="http://www.alpertron.com.ar/ECM.HTM">Computational assistance</a>

%p with(numtheory):

%p a:= proc(n) local p;

%p p:= nextprime(10^n);

%p while 1 in map(q-> 10 &^ ((p-1)/q) mod p, factorset(p-1)) or

%p 1 <> (10 &^ (p-1) mod p)

%p do p:= nextprime(p) od; p

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Feb 17 2012

%t spr10[n_]:=Module[{p=NextPrime[n]},While[PrimitiveRoot[p,10]!=10,p = NextPrime[ p]];p]; Join[{7,17},Table[spr10[10^d],{d,2,20}]] (* _Harvey P. Dale_, Nov 18 2020 *)

%K nonn

%O 0,1

%A _N. J. A. Sloane_, Feb 02 2012

%E More terms from _Alois P. Heinz_, Feb 17 2012

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Last modified April 22 09:10 EDT 2021. Contains 343174 sequences. (Running on oeis4.)