A350343 Square numbers k that are abelian orders.

1, 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1225, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 13225, 14161, 16129, 17161, 17689, 18769, 19321, 20449, 22201, 22801

Offset

1,2

Comments

k must be the square of a squarefree number. Actually, k must be the square of a cyclic number (A003277).

Number of the form (p_1*p_2*...*p_r)^2 where the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.

The smallest term with exactly n distinct prime factors is given by A350341.

From the term 25 on, no term can be divisible by 2 or 3.

Links

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

Formula

a(n) = A350342(n)^2.

Example

For primes p, p^2 is a term since every group of order p^2 is abelian. Such group is isomorphic to either C_{p^2} or C_p X C_p.
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
isA350343(n) = issquare(n) && isA051532(n)

Crossrefs

Cf. A051532 (abelian orders), A003277 (cyclic numbers), A350342, A350341.

A350152 = A350322 U A350323 is a subsequence. A350345 is the subsequence of squares of composite numbers.

Sequence in context: A179707 A247078 A077438 * A001248 A280076 A359757

Adjacent sequences: A350340 A350341 A350342 * A350344 A350345 A350346

Keyword

nonn

Author

Jianing Song, Dec 25 2021

Status

approved