A350323 Abelian orders m for which there exist at least 4 groups of order m.

1225, 4225, 5929, 7225, 13225, 14161, 15925, 17689, 20449, 20825, 23275, 25921, 28175, 34225, 34969, 43681, 45325, 46225, 47089, 48841, 50575, 55225, 57575, 61009, 64925, 67081, 70225, 70805, 71825, 72275, 77077, 80275, 82075, 89401, 89425, 92575, 93925, 96775, 97175

Offset: 1

Author

Jianing Song, Dec 25 2021

Keywords

• nonn

Comments

Abelian orders of the form (p_1)^2 * (p_2)^2 * ... * (p_r)^2 * q_1 * q_2 * ... * q_s, r >= 2, 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. Note that there are 2^r groups of such order.

No term can be divisible by 2 or 3.

Examples

			For primes p, q, if p^2 !== 1 (mod q) and q^2 !== 1 (mod p), then p^2*q^2 is a term since the 4 groups of that order are C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q}, C_{p*q} X C_{p*q}.
		

Links

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

Crossrefs

Equals A350152 \ A350322 = A051532 \ (A005117 U A060687).

Programs

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
isA350323(n) = isA051532(n) && (bigomega(n)-omega(n)>1)