A327812 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(3) (counted with multiplicity).

1, 1, 2, 1, 1, 2, 1, 2, 6, 1, 2, 2, 4, 1, 2, 2, 1, 6, 1, 2, 2, 2, 2, 4, 1, 4, 18, 2, 1, 2, 1, 2, 4, 1, 2, 6, 2, 1, 8, 4, 5, 2, 1, 2, 6, 2, 2, 4, 1, 1, 2, 4, 1, 18, 2, 4, 2, 1, 2, 4, 6, 1, 6, 2, 4, 4, 3, 2, 4, 2, 2, 12, 6, 2, 2, 2, 2, 8, 1, 8, 54, 5, 2, 4, 4, 1, 2, 4, 1, 6

Offset

1,3

Links

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

Formula

Let n = 3^e*s, gcd(3,s) = 1, then a(n) = phi(n)/ord(3,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

Example

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

Prog

(PARI) a(n) = my(s=n/3^valuation(n, 3)); eulerphi(n)/znorder(Mod(3, s))

Crossrefs

Cf. A000010.

Row 2 of A327818.

Sequence in context: A153919 A185286 A153905 * A319093 A228726 A346175

Adjacent sequences: A327809 A327810 A327811 * A327813 A327814 A327815

Keyword

nonn,easy

Author

Jianing Song, Sep 26 2019

Status

approved