A327814 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(5) (counted with multiplicity).

1, 1, 1, 2, 4, 1, 1, 2, 1, 4, 2, 2, 3, 1, 4, 2, 1, 1, 2, 8, 2, 2, 1, 4, 20, 3, 1, 2, 2, 4, 10, 2, 2, 1, 4, 2, 1, 2, 6, 8, 2, 2, 1, 4, 4, 1, 1, 4, 1, 20, 2, 6, 1, 1, 8, 4, 2, 2, 2, 8, 2, 10, 6, 2, 12, 2, 3, 2, 2, 4, 14, 4, 1, 1, 20, 4, 2, 6, 2, 8, 1, 2, 1, 4, 4, 1, 4, 4, 2, 4

Offset

1,4

Links

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

Formula

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

Mathematica

a[n_] := EulerPhi[n] / MultiplicativeOrder[5, n / 5^IntegerExponent[n, 5]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)

Prog

(PARI) a(n) = my(s=n/5^valuation(n, 5)); eulerphi(n)/znorder(Mod(5, s))

Crossrefs

Cf. A000010.

Row 4 of A327818.

Sequence in context: A010741 A094643 A094593 * A188348 A336434 A007738

Adjacent sequences: A327811 A327812 A327813 * A327815 A327816 A327817

Keyword

nonn,easy

Author

Jianing Song, Sep 26 2019

Status

approved