A350345 Squares of composite numbers k that are abelian orders.

1225, 4225, 5929, 7225, 13225, 14161, 17689, 20449, 25921, 34225, 34969, 43681, 46225, 47089, 48841, 55225, 61009, 67081, 70225, 89401, 101761, 104329, 108241, 112225, 116281, 133225, 137641, 142129, 152881, 162409, 165649, 170569, 172225, 182329, 190969

Offset

1,1

Comments

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.

Links

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

Formula

a(n) = A350344(n)^2.

Example

For primes p, q, if p^2 !== 1 (mod q), q^2 !== 1 (mod p), then p^2*q^2 is a term since every group of that order is abelian. Such group is isomorphic to C_{p^2*q^2}, C_p X C_{p*q^2}, C_q X C_{p^2*q} or 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
isA350345(n) = issquare(n) && (n>1) && !isprime(sqrtint(n)) && isA051532(n^2)

Crossrefs

Cf. A051532 (abelian orders), A050384, A350341.

Equals A350343 \ ({1} U A001248).

A350323 is a subsequence.

Sequence in context: A025405 A275256 A350323 * A183665 A250848 A014795

Adjacent sequences: A350342 A350343 A350344 * A350346 A350347 A350348

Keyword

nonn

Author

Jianing Song, Dec 25 2021

Status

approved