Cyclicity-forcing number

Definition
A natural number is termed a cyclicity-forcing number if it satisfies the following equivalent conditions:


 * 1) There exists exactly one isomorphism class of groups of that order.
 * 2) Every group of that order is a defining ingredient::cyclic group.
 * 3) Every group of that order is a direct product of cyclic Sylow subgroups.
 * 4) It is a product of distinct primes $$p_i$$ where $$p_i$$ does not divide $$p_j  - 1$$ for any two prime divisors $$p_i, p_j$$ of the order.

Equivalence of definitions
The equivalence of definitions (1)-(3) is not very hard, while the equivalence with part (4) requires some work.

Stronger properties

 * Weaker than::Prime number

Weaker properties

 * Stronger than::Square-free number
 * Stronger than::Odd number (except for the special case of the number $$2$$)
 * Stronger than::Abelianness-forcing number
 * Stronger than::Nilpotence-forcing number
 * Stronger than::Solvability-forcing number