Cyclic number (group theory)

A cyclic number^[1][2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic.^[3]

Any prime number is clearly cyclic. All cyclic numbers are square-free.^[4] Let n = p₁ p₂ … p_k where the p_i are distinct primes, then φ(n) = (p₁ − 1)(p₂ − 1)...(p_k – 1). If no p_i divides any (p_j – 1), then n and φ(n) have no common (prime) divisor, and n is cyclic.

The first cyclic numbers are 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, ... (sequence A003277 in the OEIS).