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The length of row n is delta(n) = A055034(n).

Here the smallest nonnegative complete system modulo n is used: {0,1,...,n-1}, and the reduced residue system modulo n (A038566) is the set of numbers k from this set which satisfy gcd(k, n) = 1. The present array lists only the odd numbers. For n = 1 one should take 0 because gcd(0, 1) = 1, but because 1 == 0 (mod 1) we prefer the odd 1.

This is the sub-array obtained from A038566 by deleting the even numbers.

In the multiplicative group Modd n (see a comment in A203571) each of the delta(n) members of row n forms a reduced residue class Modd n with only odd numbers. E.g., n=4 (only the positive members are listed, the negative members should be amended): [1] = {1, 7, 9, 15, 17, 23, 25, 31, 33, 39,...}; [3] = {3, 5, 11, 13, 19, 21, 27, 29, 35, 37...}. Multiplication Modd n can be done class-wise: 7*15 == 1*1 (Modd 4) = 1; 11*13 ==3*3 (Modd 4) = 1; 7*5 == 1*3 (Modd 4) = 3.

The orders 'Moddulo' n of the elements in row n are given in A216320.

The sequence of the row lengths is phi(n) = A000010 (Euler's totient).

For the notion 'reduced residue system mod n' which has, as a set, order phi(n) = A000010(n), see e.g., the Apostol reference p. 113. Here such a system with the smallest positive numbers is used. (In the Apostol reference 'order of a modulo n' is called 'exponent of a modulo n'. See the definition on p. 204.)

See A038566 where the reduced residue system mod n appears in row n.

In the chosen smallest reduced residue system mod n one can replace each element by any congruent mod n one, and the given order modulo n list will, of course, be the same. E.g., n=5, {6, -3, 13, -16} also has the orders modulo 5: 1 4 4 2, respectively.

Each order modulo n divides phi(n). See the Niven et al. reference, Corollary 2.32, p. 98.

The maximal order modulo n is given in A002322(n).

For the analog table of orders Modd n see A216320.

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 exists.

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

For the minimal polynomials C(n,x) of the algebraic number rho = 2*cos(Pi/n), n >= 1, see their coefficient table A187360. Their degree is delta(n)= phi(2*n)/2, if n >= 2, and delta(1) = 1, with Euler's totient A000010. The delta sequence is given in A055034. a(n) is the number of divisors of delta(n).

a(n) is also the number of distinct Modd n orders given in the table A216320 in row n. (For Modd n see a comment on A203571).

See the analog A062821(n), with the number of divisors of phi(n). The corresponding order table is A216327.

See a comment on A216325 on the degree delta(n) = A055034(n) of the polynomial C(n,x) of 2*cos(Pi/n) (coefficients in A187360), Here n is prime.

For p prime, delta(p) = (p - 1)/2 if p > 2 and 1 if p = 2. a(n) is the number of divisors of delta(prime(n)), with prime(n) = A000040(n).

a(n) is also the number of distinct Modd p orders, p = prime, in row prime(n) of the table A216320. (For Modd n see a comment on A203571).

See also A008328 for the mod p analog of this sequence.