Cyclic number (group theory)
- Cyclic number (group theory)
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A cyclic number[1] 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 iff any group of order n is cyclic.
Any prime number is clearly cyclic. All cyclic numbers are square-free.[2] Let n = p1 p2 … pk where the pi are distinct primes, then φ(n) = (p1 - 1)(p2 - 1)…(pk - 1). If no pi divides any (pj - 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, … (sequence A003277 in OEIS).
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