A350152 Abelian orders m for which there exist at least 2 groups with order m.

4, 9, 25, 45, 49, 99, 121, 153, 169, 175, 207, 245, 261, 289, 325, 361, 369, 423, 425, 475, 477, 529, 531, 539, 575, 637, 639, 725, 747, 765, 801, 833, 841, 845, 847, 909, 925, 931, 961, 963, 1017, 1035, 1075, 1127, 1175, 1179, 1225, 1233, 1305, 1325, 1341, 1369, 1445, 1475

Offset: 1

Author

Bernard Schott, Dec 18 2021

Keywords

• nonn

Comments

This sequence lists the abelian orders when there is an abelian group that is distinct from cyclic group. When there is only one group of order k, then k is in A003277 and this group is the cyclic group C_k.

Except for a(1) = 4, all the terms are odd, because of the existence of a non-abelian dihedral group D_{2*n} of order 2*n for each n > 2.

Every p^2, p prime, is a term and the 2 corresponding abelian groups are C_{p^2} and C_p X C_p.

Examples

			4 is a term because the 2 groups of order 4 that are C_4 and C_2 X C_2, the Klein four-group, are both abelian and a(1) = 4 because there is no smallest order with 2 abelian groups.
45 is a term because the 2 groups of order 45 that are C_45 and C_5  X C_3 X C_3 are both abelian.
99 is another term because the 2 groups of order 99 that are C_99 and C_11 X C_3 X C_3 are both abelian.
		

Links

• Mathematics Stack Exchange, Group of order 45 is abelian.

Crossrefs

Equals A051532 \ A003277.

Cf. A000001, A000688.

A001248 is a subsequence.

Programs

Mathematica

f[p_, e_] := Product[1 - p^i, {i, 1, e}]; q[n_] := !CoprimeQ[EulerPhi[n], n] && Module[{fct = FactorInteger[n], e}, e = fct[[;; , 2]]; Max[e] < 3 && CoprimeQ[Abs[Times @@ f @@@ fct], n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 18 2021 *)

Formula

m such that A000001(m) = A000688(m) > 1.

Extensions

More terms from Michel Marcus, Dec 18 2021