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

%I A350152 #36 Mar 31 2023 09:18:02
%S A350152 4,9,25,45,49,99,121,153,169,175,207,245,261,289,325,361,369,423,425,
%T A350152 475,477,529,531,539,575,637,639,725,747,765,801,833,841,845,847,909,
%U A350152 925,931,961,963,1017,1035,1075,1127,1175,1179,1225,1233,1305,1325,1341,1369,1445,1475
%N A350152 Abelian orders m for which there exist at least 2 groups with order m.
%C A350152 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.
%C A350152 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.
%C A350152 Every p^2, p prime, is a term and the 2 corresponding abelian groups are C_{p^2} and C_p X C_p.
%H A350152 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3719529/group-of-order-45-is-abelian">Group of order 45 is abelian</a>.
%F A350152 m such that A000001(m) = A000688(m) > 1.
%e A350152 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.
%e A350152 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.
%e A350152 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.
%t A350152 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 *)
%Y A350152 Equals A051532 \ A003277.
%Y A350152 Cf. A000001, A000688.
%Y A350152 A001248 is a subsequence.
%K A350152 nonn
%O A350152 1,1
%A A350152 _Bernard Schott_, Dec 18 2021
%E A350152 More terms from _Michel Marcus_, Dec 18 2021