A327813 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(4) (counted with multiplicity).

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 8, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 2, 4, 2, 4, 6, 16, 4, 4, 4, 4, 2, 2, 4, 8, 4, 4, 6, 4, 4, 2, 2, 16, 2, 2, 8, 4, 2, 2, 4, 8, 4, 2, 2, 8, 2, 6, 12, 32, 8, 4, 2, 8, 4, 4, 2, 8, 8, 2, 4, 4, 4, 4, 2, 16, 2, 4, 2, 8, 16, 6, 4, 8, 8, 4

Offset

1,3

Links

Table of n, a(n) for n=1..90.

Formula

Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(4,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

Example

Let GF(4) = GF(2)[w], where w^2 + w + 1 = 0. Factorizations of the n-th cyclotomic polynomial over GF(4) for n <= 10:
n = 1: x + 1;
n = 2: x + 1;
n = 3: (x + w)*(x + (w+1));
n = 4: (x + 1)^2;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: (x + w)*(x + (w+1));
n = 7: (x^3 + x + 1)*(x^3 + x^2 + 1);
n = 8: (x + 1)^4;
n = 9: (x^3 + w)*(x^3 + (w+1));
n = 10: x^4 + x^3 + x^2 + x + 1.

Prog

(PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(4, s))

Crossrefs

Cf. A000010.

Row 3 of A327818.

Sequence in context: A076500 A344887 A060594 * A104361 A211449 A086876

Adjacent sequences: A327810 A327811 A327812 * A327814 A327815 A327816

Keyword

nonn,easy

Author

Jianing Song, Sep 26 2019

Status

approved