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

%I A350322 #26 Aug 01 2025 11:09:37
%S A350322 4,9,25,45,49,99,121,153,169,175,207,245,261,289,325,361,369,423,425,
%T A350322 475,477,529,531,539,575,637,639,725,747,765,801,833,841,845,847,909,
%U A350322 925,931,961,963,1017,1035,1075,1127,1175,1179,1233,1305,1325,1341,1369,1445,1475
%N A350322 Abelian orders m for which there exist exactly 2 groups of order m.
%C A350322 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.
%C A350322 Note that the smallest abelian order with precisely 2^n groups must be the square of a squarefree number.
%C A350322 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.
%H A350322 Jianing Song, <a href="/A350322/b350322.txt">Table of n, a(n) for n = 1..10000</a>
%e A350322 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.
%e A350322 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}.
%o A350322 (PARI) isA054395(n) = {
%o A350322   my(p=gcd(n, eulerphi(n)), f);
%o A350322   if (!isprime(p), return(0));
%o A350322   if (n%p^2 == 0, return(1 == gcd(p+1, n)));
%o A350322   f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k, 1]%p==1);
%o A350322 } \\ _Gheorghe Coserea_'s program for A054395
%o A350322 isA350322(n) = isA054395(n) && (bigomega(n)-omega(n)==1)
%o A350322 (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
%o A350322 isA350322(n) = isA051532(n) && (bigomega(n)-omega(n)==1)
%o A350322 (Python)
%o A350322 def is_ok(m):
%o A350322     f = factorint(m)
%o A350322     return (
%o A350322         sum(f.values()) == len(f) + 1 and
%o A350322         all((q - 1) % p > 0 for p in f for q in f) and
%o A350322         (m := next(p for p, e in f.items() if e == 2) ** 2 - 1) and
%o A350322         all(m % q > 0 for q in f)) # _David Radcliffe_, Jul 30 2025
%Y A350322 Equals A060687 INTERSECT A051532 = A054395 INTERSECT A051532 = A054395 INTERSECT A060687 = A054395 INTERSECT A013929.
%Y A350322 Equals A350152 \ A350323.
%Y A350322 Equals A054395 \ A350586.
%Y A350322 Subsequence of A350152.
%Y A350322 A001248 and A350332 are subsequences.
%K A350322 nonn
%O A350322 1,1
%A A350322 _Jianing Song_, Dec 25 2021