A103310 Largest prime primitive root of n that is less than n, or 0 if none exists.

0, 0, 0, 2, 3, 3, 5, 5, 0, 5, 7, 7, 0, 11, 5, 0, 0, 11, 11, 13, 0, 0, 19, 19, 0, 23, 19, 23, 0, 19, 0, 17, 0, 0, 31, 0, 0, 19, 29, 0, 0, 29, 0, 29, 0, 0, 43, 43, 0, 47, 47, 0, 0, 41, 47, 0, 0, 0, 47, 47, 0, 59, 53, 0, 0, 0, 0, 61, 0, 0, 0, 67, 0, 59, 61, 0, 0, 0, 0, 59, 0, 59, 71, 79, 0, 0, 73

Offset: 0

Author

Harry J. Smith, Jan 29 2005

Keywords

• easy
• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Primitive Root.

Crossrefs

Cf. A001918, A046144, A046145, A046146, A103309.

Programs

Maple

hasproot:= proc(n)
  if n::odd then nops(numtheory:-factorset(n))=1
  else padic:-ordp(n,2)=1 and nops(numtheory:-factorset(n/2))=1
  fi
end proc;
hasproot(2):= true: hasproot(4):= true:
f:= proc(n) local p,t;
  if not hasproot(n) then return 0 fi;
  t:= numtheory:-phi(n);
  p:= prevprime(n);
  while not numtheory:-order(p,n)=t do
    if p = 2 then return 0 fi;
    p:= prevprime(p);
  od;
  p
end proc:
f(0):= 0: f(1):= 0: f(2):= 0:
map(f, [$0..100]); # Robert Israel, Sep 08 2020

Mathematica

a[n_] := Module[{R = PrimitiveRootList[n], s}, s = Select[R, # < n && PrimeQ[#]&]; If[s == {}, 0, s[[-1]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 01 2023 *)