cp's OEIS Frontend

If m is an abelian order, then m = (p_1)^2 * (p_2)^2 * ... * (p_r)^2 * q_1 * q_2 * ... * q_s, where p_1, p_2, ... p_r, q_1, q_2, ..., q_s are distinct primes such that (p_i)^2 !== 1 (mod p_j) for i != j, (p_i)^2 !== 1 (mod q_j), q_i !== 1 (mod p_j), q_i !== 1 (mod q_j) for i != j. In this case there are 2^r groups of order m.

Note that the smallest abelian order with precisely 2^n groups must be the square of a squarefree number.

a(n) is the smallest k with n distinct prime factors such that k^2 is an abelian order.

a(n) is the smallest number of the form p_1*p_2*...*p_n where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.

a(n) exists for all n.

Except for a(1) = 2, no term can be divisible by 2 or 3. Conjecture: lpf(a(n+1)) >= lpf(a(n)) for all n, where lpf = least prime factor. - David A. Corneth and Jianing Song, Jan 03 2022

k must be the square of a squarefree number. Actually, k must be the square of a cyclic number (A003277).

Number of the form (p_1*p_2*...*p_r)^2 where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.

The smallest term with exactly n distinct prime factors is given by A350341.

From the term 25 on, no term can be divisible by 2 or 3.

Square numbers k that are abelian orders with at least 4 groups.

Number of the form (p_1*p_2*...*p_r)^2 where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.

The smallest square number k that is an abelian order with at least 8 groups is A350341(3) = 354025.

No term can be divisible by 2 or 3.