A046144 Number of primitive roots modulo n.

1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 4, 0, 4, 2, 0, 0, 8, 2, 6, 0, 0, 4, 10, 0, 8, 4, 6, 0, 12, 0, 8, 0, 0, 8, 0, 0, 12, 6, 0, 0, 16, 0, 12, 0, 0, 10, 22, 0, 12, 8, 0, 0, 24, 6, 0, 0, 0, 12, 28, 0, 16, 8, 0, 0, 0, 0, 20, 0, 0, 0, 24, 0, 24, 12, 0, 0, 0, 0, 24, 0, 18, 16, 40, 0, 0, 12, 0, 0, 40, 0, 0

Offset

1,5

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math/0605019 [math.NT], 2006-2017.

Eric Weisstein's World of Mathematics, Primitive Root.

Formula

a(n) is equal to A010554(n) unless n is a term of A033949, in which case a(n)=0.

Maple

A046144 := proc(n)
    local a, eulphi, m;
    if n = 1 then
        return 1;
    end if;
    eulphi := numtheory[phi](n) ;
    a := 0 ;
    for m from 0 to n-1 do
        if numtheory[order](m, n) = eulphi then
            a := a + 1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Jan 12 2016

Mathematica

Prepend[ Table[ If[ IntegerQ[ PrimitiveRoot[n]] , EulerPhi[ EulerPhi[n]], 0], {n, 2, 91}], 1] (* Jean-François Alcover, Sep 13 2011 *)

Prog

(PARI) for(i=1, 100, p=0; for(q=1, i, if(gcd(q, i)==1 && znorder(Mod(q, i)) == eulerphi(i), p++)); print1(p, ", ")) /* V. Raman, Nov 22 2012 */
(PARI) a(n) = my(s=znstar(n)); if(#(s.cyc)>1, 0, eulerphi(s.no)) \\ Jeppe Stig Nielsen, Oct 18 2019
(Perl) use ntheory ":all"; my @A = map { !defined znprimroot($_) ? 0 : euler_phi(euler_phi($_)); } 0..10000; say "$_ $A[$_]" for 1..$#A; # Dana Jacobsen, Apr 28 2017

Crossrefs

Cf. A001918, A010554, A033949, A046145, A046146, A008330, A002233, A071894, A219027.

Sequence in context: A117448 A093321 A302015 * A335904 A144736 A137423

Adjacent sequences: A046141 A046142 A046143 * A046145 A046146 A046147

Keyword

nonn

Author

Eric W. Weisstein

Status

approved