A072209 Number of primitive roots of those integers with at least one primitive root.

1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 2, 8, 2, 6, 4, 10, 8, 4, 6, 12, 8, 8, 12, 6, 16, 12, 10, 22, 12, 8, 24, 6, 12, 28, 16, 8, 20, 24, 24, 12, 24, 18, 16, 40, 12, 40, 22, 32, 12, 40, 32, 24, 52, 36, 48, 28, 40, 16, 40, 36, 48, 20, 64, 44, 24, 24, 72, 40, 48, 24, 18, 54

Offset: 1

Author

Lekraj Beedassy, Jul 03 2002

Keywords

• nice
• nonn

Comments

Essentially sequence A046144 with all zero entries deleted.

Links

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

Crossrefs

Cf. A033948, A046144.

Programs

Mathematica

Reap[ Do[ If[n == 1, Sow[1], If[ IntegerQ[ PrimitiveRoot[n]], Sow[ EulerPhi[ EulerPhi[n]]]]] , {n, 1, 100}]][[2, 1]] (* Jean-François Alcover, Feb 24 2012 *)
Join[{1},(Length/@PrimitiveRootList[Range[300]])/.(0->Nothing)] (* Harvey P. Dale, Oct 01 2024 *)

PARI

is(n)=if(n%2, isprimepower(n) || n==1, n==2 || n==4 || (isprimepower(n/2, &n) && n>2));
lista(nn) = for (n=1, nn, if (is(n), print1(eulerphi(eulerphi(n)), ", "))); \\ Michel Marcus, May 12 2017

Python

from sympy import primepi, integer_nthroot, totient
def A072209(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n-1+x-(x>=2)-(x>=4)-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length()))-sum(primepi(integer_nthroot(x>>1,k)[0])-1 for k in range(1,x.bit_length()-1)))
    return totient(totient(bisection(f,n,n))) # Chai Wah Wu, Feb 24 2025

Formula

a(n) = phi(phi(A033948(n))).