A327815 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(7) (counted with multiplicity).

1, 1, 2, 1, 1, 2, 6, 2, 2, 1, 1, 2, 1, 6, 2, 4, 1, 2, 6, 2, 12, 1, 1, 4, 5, 1, 2, 6, 4, 2, 2, 4, 2, 1, 6, 2, 4, 6, 2, 4, 1, 12, 7, 2, 2, 1, 2, 8, 42, 5, 2, 2, 2, 2, 2, 12, 12, 4, 2, 4, 1, 2, 12, 4, 4, 2, 1, 2, 2, 6, 1, 4, 3, 4, 10, 6, 6, 2, 1, 8, 2, 1, 2, 12, 4, 7, 8, 4, 1, 2

Offset

1,3

Links

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

Formula

Let n = 7^e*s, gcd(7,s) = 1, then a(n) = phi(n)/ord(7,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(7) for n <= 10:
n = 1: x - 1;
n = 2: x + 1;
n = 3: (x + 3)*(x - 2);
n = 4: x^2 + 1;
n = 5: x^4 + x^3 + x^2 + x + 1;
n = 6: (x + 2)*(x - 3);
n = 7: (x - 1)^6;
n = 8: (x^2 + 3x + 1)*(x^2 - 3x + 1);
n = 9: (x^3 + 3)*(x^3 - 2);
n = 10: x^4 - x^3 + x^2 - x^2 + 1.

Prog

(PARI) a(n) = my(s=n/7^valuation(n, 7)); eulerphi(n)/znorder(Mod(7, s))

Crossrefs

Cf. A000010.

Row 5 of A327818.

Sequence in context: A206497 A123218 A298608 * A007736 A107042 A134830

Adjacent sequences: A327812 A327813 A327814 * A327816 A327817 A327818

Keyword

nonn,easy

Author

Jianing Song, Sep 26 2019

Status

approved