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A350152
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Abelian orders m for which there exist at least 2 groups with order m.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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