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A023049 Smallest prime > n having primitive root n, or 0 if no such prime exists.

%I A023049 #24 Nov 11 2024 22:25:19
%S A023049 2,3,5,0,7,11,11,11,0,17,13,17,19,17,19,0,23,29,23,23,23,31,47,31,0,
%T A023049 29,29,41,41,41,47,37,43,41,37,0,59,47,47,47,47,59,47,47,47,67,59,53,
%U A023049 0,53,53,59,71,59,59,59,67,73,61,73,67,71,67,0,71,79,71,71,71,79,83,83,83,79
%N A023049 Smallest prime > n having primitive root n, or 0 if no such prime exists.
%C A023049 Indices of record values of a(n)-n are (1, 2, 3, 6, 10, 18, 23, 78, 102, 105, 488, 652, 925, ...). Record values of a(n)/n are 3/2, 5/3, 11/6, 47/23, ... (Is there another n with a(n) > 2n ?) - _M. F. Hasler_, Feb 21 2017
%H A023049 Robert Israel, <a href="/A023049/b023049.txt">Table of n, a(n) for n = 1..10000</a>
%F A023049 a(n) = 0 iff n is a square > 1. - _M. F. Hasler_, Feb 21 2017
%p A023049 f:= proc(n) local p;
%p A023049   if issqr(n) then return 0 fi;
%p A023049   p:= nextprime(n);
%p A023049   do
%p A023049     if numtheory:-order(n,p) = p-1 then return p fi;
%p A023049     p:= nextprime(p);
%p A023049   od
%p A023049 end proc:
%p A023049 f(1):= 2:
%p A023049 map(f, [$1..100]); # _Robert Israel_, Feb 21 2017
%t A023049 a[n_] := For[p = 2, p <= 2 n + 1, p = NextPrime[p], If[MemberQ[ PrimitiveRootList[p], n], Return[p]]] /. Null -> 0; Array[a, 100] (* _Jean-François Alcover_, Mar 05 2019 *)
%o A023049 (PARI) A023049(n)={issquare(n)||forprime(p=n+1,,znorder(Mod(n,p))==p-1&&return(p));(n==1)*2} \\ _M. F. Hasler_, Feb 21 2017
%Y A023049 See also A056619, where the primitive root may be larger than the prime, whereas in A023049 it may not be.
%Y A023049 Cf. A377938, A377939.
%K A023049 nonn
%O A023049 1,1
%A A023049 _David W. Wilson_