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A206552
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Moduli n for which the multiplicative group Modd n is non-cyclic (acyclic).
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5
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12, 20, 24, 28, 30, 36, 40, 42, 44, 48, 52, 56, 60, 63, 65, 66, 68, 70, 72, 76, 78, 80, 84, 85, 88, 90, 91, 92, 96, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 133, 136, 138, 140, 144, 145, 148, 150, 152, 154, 156, 160, 164, 165
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OFFSET
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1,1
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COMMENTS
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For Modd n (not to be confused with mod n) see a comment on A203571.
Precisely these numbers n (only the ones <=165 are shown above) have no primitive root Modd n. See the zero entries of A206550, except A206550(1) = 0 which stands for a primitive root 0.
The multiplicative Modd n group is the Galois group Gal(Q(rho(n))/Q), with the algebraic number rho(n) := 2*cos(Pi/n) with minimal polynomial C(n,x), whose coefficients are given in A187360.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 12 because A206550(12) = 0 for the first time, not counting A206550(1) = 0. The cycle structure of the multiplicative Modd 12 group is [[5,1],[7,1],[11,1]]. This is the (abelian, non-cyclic) group Z_2 x Z_2 (isomorphic to the Klein group V_4 or Dih_2).
a(2) = 20 because A206550(20) = 0 for the second time, not counting A206550(1) = 0. The cycle structure of the multiplicative Modd 20 group is [[3,9,13,1],[7,9,17,1],[11,1],[19,1]]. This is the (abelian, non-cyclic) group Z_4 x Z_2.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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