login
A327817
Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(9) (counted with multiplicity).
2
1, 1, 2, 2, 2, 2, 2, 4, 6, 2, 2, 4, 4, 2, 4, 4, 2, 6, 2, 4, 4, 2, 2, 8, 2, 4, 18, 4, 2, 4, 2, 4, 4, 2, 4, 12, 4, 2, 8, 8, 10, 4, 2, 4, 12, 2, 2, 8, 2, 2, 4, 8, 2, 18, 4, 8, 4, 2, 2, 8, 12, 2, 12, 4, 8, 4, 6, 4, 4, 4, 2, 24, 12, 4, 4, 4, 4, 8, 2, 16, 54, 10, 2, 8, 8, 2, 4, 8, 2, 12
OFFSET
1,3
LINKS
FORMULA
Let n = 3^e*s, gcd(3,s) = 1, then a(n) = phi(n)/ord(9,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
EXAMPLE
Let GF(9) = GF(3)[i], where i^2 = -1. Factorizations of the n-th cyclotomic polynomial over GF(9) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: (x - 1)^2;
n = 4: (x + i)*(x - i);
n = 5: (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
n = 6: (x + 1)^2;
n = 7: (x^3 + (-1+i)*x^2 + (1+i)*x - 1)*(x^3 + (-1-i)*x^2 + (1-i)*x - 1);
n = 8: (x + (1+i))*(x + (1-i))*(x + (-1+i))*(x + (-1-i));
n = 9: (x - 1)^6;
n = 10: (x^2 + (1+i)*x + 1)*(x^2 + (1-i)*x + 1).
MATHEMATICA
a[n_] := EulerPhi[n] / MultiplicativeOrder[9, n / 3^IntegerExponent[n, 3]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)
PROG
(PARI) a(n) = my(s=n/3^valuation(n, 3)); eulerphi(n)/znorder(Mod(9, s))
CROSSREFS
Cf. A000010.
Row 7 of A327818.
Sequence in context: A083499 A029103 A175732 * A353643 A168260 A008737
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Sep 26 2019
STATUS
approved