A350322 Abelian orders m for which there exist exactly 2 groups of order m.

4, 9, 25, 45, 49, 99, 121, 153, 169, 175, 207, 245, 261, 289, 325, 361, 369, 423, 425, 475, 477, 529, 531, 539, 575, 637, 639, 725, 747, 765, 801, 833, 841, 845, 847, 909, 925, 931, 961, 963, 1017, 1035, 1075, 1127, 1175, 1179, 1233, 1305, 1325, 1341, 1369, 1445, 1475

Offset: 1

Author

Jianing Song, Dec 25 2021

Keywords

• nonn

Comments

Abelian orders of the form p^2 * q_1 * q_2 * ... * q_s, where p, q_1, q_2, ..., q_s are distinct primes such that p^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.

Except for a(1) = 4, all terms are odd. The terms that are divisible by 3 are of the form 9 * q_1 * q_2 * ... * q_s, where q_i are distinct primes congruent to 5 modulo 6, q_i !== 1 (mod q_j) for i != j.

Examples

			For primes p, p^2 is a term since the 2 groups of that order are C_{p^2} and C_p X C_p.
For primes p, q, if p^2 !== 1 (mod q) and q !== 1 (mod p), then p^2*q is a term since the 2 groups of that order are C_{p^2*q} and C_p X C_{p*q}.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..10000

Crossrefs

Equals A060687 INTERSECT A051532 = A054395 INTERSECT A051532 = A054395 INTERSECT A060687 = A054395 INTERSECT A013929.

Equals A350152 \ A350323.

Equals A054395 \ A350586.

Subsequence of A350152.

A001248 and A350332 are subsequences.

Programs

PARI

isA054395(n) = {
  my(p=gcd(n, eulerphi(n)), f);
  if (!isprime(p), return(0));
  if (n%p^2 == 0, return(1 == gcd(p+1, n)));
  f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k, 1]%p==1);
} \\ Gheorghe Coserea's program for A054395
isA350322(n) = isA054395(n) && (bigomega(n)-omega(n)==1)

PARI

isA051532(n) = my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(0), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV's program for A051532
isA350322(n) = isA051532(n) && (bigomega(n)-omega(n)==1)

Python

def is_ok(m):
    f = factorint(m)
    return (
        sum(f.values()) == len(f) + 1 and
        all((q - 1) % p > 0 for p in f for q in f) and
        (m := next(p for p, e in f.items() if e == 2) ** 2 - 1) and
        all(m % q > 0 for q in f)) # David Radcliffe, Jul 30 2025