A219029 -id:A219029 - cp's OEIS frontend

A219027 Number of non-primitive roots for n, less than n.

0, 0, 1, 2, 2, 4, 4, 7, 6, 7, 6, 11, 8, 11, 14, 15, 8, 15, 12, 19, 20, 17, 12, 23, 16, 21, 20, 27, 16, 29, 22, 31, 32, 25, 34, 35, 24, 31, 38, 39, 24, 41, 30, 43, 44, 35, 24, 47, 36, 41, 50, 51, 28, 47, 54, 55, 56, 45, 30, 59, 44, 53, 62, 63, 64, 65, 46, 67, 68

Offset: 1

V. Raman, Nov 10 2012

nonn

a(n) will be the same as A219029(n) except when n is a member of A033949 or n = 1, i.e. n is not 2, 4, prime, power of a prime, twice a prime, or twice a prime power. In such cases, when n is a member of A033949, then a(n) = n-1.

Cf. A008330 (number of primitive roots for the n-th prime, less than n-th prime).
Cf. A046144 (number of primitive roots for n, less than n).
Cf. A010554 (value of phi(phi(n))).
Cf. A219029.

for(i=1,100,p=0;for(q=1,i-1,if(gcd(q,i)>1||znorder(Mod(q,i))!=eulerphi(i),p++));print1(p","))

n-1-A046144(n).

A219428 a(n) = n - 1 - phi(n).

Original entry on oeis.org

-1, 0, 0, 1, 0, 3, 0, 3, 2, 5, 0, 7, 0, 7, 6, 7, 0, 11, 0, 11, 8, 11, 0, 15, 4, 13, 8, 15, 0, 21, 0, 15, 12, 17, 10, 23, 0, 19, 14, 23, 0, 29, 0, 23, 20, 23, 0, 31, 6, 29, 18, 27, 0, 35, 14, 31, 20, 29, 0, 43, 0, 31, 26, 31, 16, 45, 0, 35, 24, 45, 0, 47

Offset: 1

V. Raman, Nov 20 2012

Apart from the first term, the same as A016035.
For n > 1, a(n) is also the number of numbers below n which are not coprime to n.
a(n) = 0 if n is prime.
x^(n - 1 - phi(n)) is congruent to x^(n - 1) mod n, if x is coprime to n, since x^phi(n) is congruent to 1 (mod n) if x is coprime to n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A219029.

[(n - 1 - (EulerPhi(n))): n in [1..100]]; // Vincenzo Librandi, Jan 26 2013

Table[n - (EulerPhi[n] + 1), {n, 75}] (* Alonso del Arte, Nov 17 2012 *)

for(n=1,100,print1(n-1-eulerphi(n)","))

a(n) = n - 1 - A000010(n) = A051953(n) - 1 = cototient(n) - 1. - Omar E. Pol, Nov 21 2012

cp's OEIS Frontend

A219027 Number of non-primitive roots for n, less than n.

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