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A350323
Abelian orders m for which there exist at least 4 groups of order m.
4
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
1,1
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.
LINKS
EXAMPLE
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}.
PROG
(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)
CROSSREFS
Sequence in context: A025407 A025405 A275256 * A350345 A183665 A250848
KEYWORD
nonn
AUTHOR
Jianing Song, Dec 25 2021
STATUS
approved