A350342 -id:A350342 - cp's OEIS frontend

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

Jianing Song, Dec 25 2021

nonn

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.

			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}.

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

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.

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)

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

A350344 Composite k such that k^2 is an abelian order.

Original entry on oeis.org

35, 65, 77, 85, 115, 119, 133, 143, 161, 185, 187, 209, 215, 217, 221, 235, 247, 259, 265, 299, 319, 323, 329, 335, 341, 365, 371, 377, 391, 403, 407, 413, 415, 427, 437, 451, 469, 481, 485, 493, 511, 515, 517, 527, 533, 535, 551, 553, 559, 565, 583, 589, 595

Offset: 1

Jianing Song, Dec 25 2021

nonn

Numbers k such that k^2 is an abelian order with at least 4 groups.
Number of the form p_1*p_2*...*p_r where r > 1, the p_i are distinct primes and no (p_j)^2-1 is divisible by any p_i.
The smallest number k such that k^2 is an abelian order with at least 8 groups is A350340(3) = 595.
No term can be divisible by 2 or 3.

			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}.
95 is not a term since 95^2 = 5^2 * 19^2 is not an abelian order. Note that 95 itself is a cyclic number.

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

Cf. A051532 (abelian orders), A050384, A350340.

Equals A350342 \ ({1} U A000040).

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
isA350344(n) = (n>1) && !isprime(n) && isA051532(n^2)

A350345(n) = a(n)^2.

cp's OEIS Frontend

A350343 Square numbers k that are abelian orders.

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