A350342 Numbers k such that k^2 is an abelian order.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 119, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 173, 179, 181, 185, 187, 191, 193, 197, 199, 209

Offset

1,2

Comments

Such k must be squarefree. Actually, such k must be a cyclic number (A003277).

Number of the form p_1*p_2*...*p_r 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 A350340.

From the term 5 on, no term can be divisible by 2 or 3.

Links

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

Formula

A350343(n) = a(n)^2.

Example

For primes p, p 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*q 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
isA350342(n) = isA051532(n^2)

Crossrefs

Cf. A051532 (abelian orders), A003277 (cyclic numbers), A350343, A350340.

A350344 is the subsequence of composite numbers.

Sequence in context: A089063 A339819 A247142 * A369093 A171019 A249825

Adjacent sequences: A350339 A350340 A350341 * A350343 A350344 A350345

Keyword

nonn

Author

Jianing Song, Dec 25 2021

Status

approved