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If n belongs to A206551 (cyclic multiplicative group Modd n) then there exist precisely a(n) primitive roots Modd n. For these n values the number of entries in row n of the table A216319 with value delta(n) (the row length) is a(n). Note that a(n) is also defined for the complementary n values from A206552 (non-cyclic multiplicative group Modd n) for which no primitive root Modd n exitsexists.

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(PARI) a(n)=eulerphi(ceil(eulerphi(2*n)/2)) \\ Charles R Greathouse IV, Feb 21 2013

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allocated for Wolfdieter Langphi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 (degree of minimal polynomials with coefficients given in A187360).

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 6, 4, 2, 4, 10, 4, 4, 4, 6, 4, 6, 4, 8, 8, 4, 8, 4, 4, 6, 6, 4, 8, 8, 4, 12, 8, 4, 10, 22, 8, 12, 8, 8, 8, 12, 6, 8, 8, 6, 12, 28, 8, 8, 8, 6, 16, 8, 8, 20, 16, 10, 8, 24, 8, 12, 12, 8, 12, 8, 8, 24, 16, 18, 16, 40, 8, 16, 12

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If n belongs to A206551 (cyclic multiplicative group Modd n) then there exist precisely a(n) primitive roots Modd n. For these n values the number of entries in row n of the table A216319 with value delta(n) (the row length) is a(n). Note that a(n) is also defined for the complementary n values from A206552 (non-cyclic multiplicative group Modd n) for which no primitive root Modd n exits.

See also A216322 for the number of primitive roots Modd n.

a(n) = phi(delta(n)), n >= 1, with phi = A000010 (Euler's totient) and delta = A055034 with delta(1) = 1 and delta(n) = phi(2*n)/2 if n >= 2.

a(8) = 2 because delta(8) = 4 and phi(4) = 2. There are 2 primitive roots Modd 8, namely 3 and 5 (see the two 4s in row n=8 of A216320). 8 = A206551(8).

a(12) = 2 because delta(12) = 4 and phi(4) = 2. But there is no primitive root Modd 12, because 4 does not show up in row n=12 of A216320. 12 = A206552(1).

Cf. A000010, A055034, A216319, A216320, A216322, A010554 (analog in modulo n case).

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Wolfdieter Lang, Sep 21 2012

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allocated for Wolfdieter Lang

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