A216321 -id:A216321 - cp's OEIS frontend

A216322 Number of primitive roots Modd n (see A216321).

1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 0, 2, 2, 2, 4, 4, 2, 6, 0, 2, 4, 10, 0, 4, 4, 6, 0, 6, 0, 8, 8, 4, 8, 4, 0, 6, 6, 4, 0, 8, 0, 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

Offset: 1

Wolfdieter Lang, Sep 21 2012

nonn

This sequence coincides with A216321 for all n values from A206551 (cyclic multiplicative Modd n group) and the entry is 0 otherwise (if no primitive root exists, that is, n is from the complementary sequence A206552).

			a(8) = phi(phi(2*8)/2) = 2 , with phi = A000010, because 8 = A206551(8).
a(12) = 0 because 12 = A206552(1).

Cf. A216321, A046144 (modulo n analog).

a(n) = A216321(n) if n belongs to the sequence A206551 and a(n)=0 if n belongs to A206552.

A216320 Irregular triangle: row n lists the Modd n order of the odd members of the reduced smallest nonnegative residue class modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 4, 4, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 5, 5, 5, 1, 2, 2, 2, 1, 3, 2, 6, 3, 6, 1, 3, 6, 3, 6, 2, 1, 4, 2, 4, 1, 8, 8, 4, 4, 8, 8, 2, 1, 8, 8, 8, 4, 8, 2, 4, 1, 6, 6, 3, 3, 2, 1, 9, 9, 3, 9, 3, 9, 9, 9, 1, 4, 4, 2, 2, 4, 4, 2, 1, 3, 6, 2, 3, 6

Offset: 1

Wolfdieter Lang, Sep 21 2012

The length of row n is delta(n):=A055034(n).
For the multiplicative group Modd n see a comment on A203571, and also on A216319.
A216319(n,k)^a(n,k) == +1 (Modd n), n >= 1.
If the Modd n order of an (odd) element from row n of A216319 is delta(n) (the row length) then this element is a primitive root Modd n. There is no primitive root Modd n if no such element of order delta(n) exists. For example, n = 12, 20, ... (see A206552 for more of these n values). There are phi(delta(n)) = A216321(n) such primitive roots Modd n if there exists one, where phi=A000010 (Euler's totient). The multiplicative group Modd n is cyclic if and only if there exists a primitive root Modd n. The multiplicative group Modd n is isomorphic to the Galois group G(Q(rho(n))/Q) with the algebraic number rho(n) := 2*cos(Pi/n), n>=1.

			The table a(n,k) begins:
  n\k 1  2  3  4  5  6  7  8  9 ...
  1   1
  2   1
  3   1
  4   1  2
  5   1  2
  6   1  2
  7   1  3  3
  8   1  4  4  2
  9   1  3  3
  10  1  4  4  2
  11  1  5  5  5  5
  12  1  2  2  2
  13  1  3  2  6  3  6
  14  1  3  6  3  6  2
  15  1  4  2  4
  16  1  8  8  4  4  8  8  2
  17  1  8  8  8  4  8  2  4
  18  1  6  6  3  3  2
  19  1  9  9  3  9  3  9  9  9
  20  1  4  4  2  2  4  4  2
  ...
a(7,2) = 3 because A216319(7,2) = 3 and 3^1 == 3 (Modd 7);
  3^2 = 9 == 5 (Modd 7) because floor(9/7)= 1 which is odd, therefore 9 (Modd 7) = -9 (mod 7) = 5; 3^3 == 5*3 (Modd n)
  = +1 because floor(15/7)=2 which is even, therefore 15 (Modd 7) = 15 (modd 7) = +1.
Row n=12 is the first row without an order = delta(n) (row length), in this case 4. Therefore there is no primitive root Modd 12, and the multiplicative group Modd 12 is non-cyclic.
  Its cycle structure is [[5,1],[7,1],[11,1]] which is the group Z_2 x Z_2 (the Klein 4-group).

Cf. A203571, A216321.

a(n,k) = order of A216319(n,k) Modd n, n>=1, k=1, 2, ..., A055034(n). This means: A216319(n,k)^a(n,k) == +1 (Modd n), n>=1, and a(n,k) is the smallest positive integer exponent satisfying this congruence. For Modd n see a comment on A203571.

cp's OEIS Frontend

A216322 Number of primitive roots Modd n (see A216321).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula