A327816 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(8) (counted with multiplicity).

1, 1, 1, 2, 1, 1, 6, 4, 3, 1, 1, 2, 3, 6, 2, 8, 2, 3, 3, 2, 6, 1, 2, 4, 1, 3, 3, 12, 1, 2, 6, 16, 2, 2, 6, 6, 3, 3, 6, 4, 2, 6, 3, 2, 6, 2, 2, 8, 6, 1, 4, 6, 1, 3, 2, 24, 6, 1, 1, 4, 3, 6, 18, 32, 12, 2, 3, 4, 2, 6, 2, 12, 24, 3, 2, 6, 6, 6, 6, 8, 3, 2, 1, 12, 8, 3, 2, 4, 8, 6

Offset

1,4

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(8,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

Example

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

Mathematica

a[n_] := EulerPhi[n] / MultiplicativeOrder[8, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)

Prog

(PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(8, s))

Crossrefs

Cf. A000010.

Row 6 of A327818.

Sequence in context: A186023 A103880 A135899 * A047920 A350269 A249673

Adjacent sequences: A327813 A327814 A327815 * A327817 A327818 A327819

Keyword

nonn,easy

Author

Jianing Song, Sep 26 2019

Status

approved