A cyclic number is a natural number n such that n and ÃÂ(n) are coprime. Here ÃÂ 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.
Any prime number is clearly cyclic. All cyclic numbers are square-free. Let , where the p<sub>i</sub> are distinct primes, then . If no p<sub>i</sub> divides any , then n and ÃÂ(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, ... .