

A327816


Number of irreducible factors in the factorization of the nth cyclotomic polynomial over GF(8) (counted with multiplicity).


1



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
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OFFSET

1,4


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(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 nth 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.


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 A249673 A144655
Adjacent sequences: A327813 A327814 A327815 * A327817 A327818 A327819


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Sep 26 2019


STATUS

approved



