A327813 - OEIS

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A327813

Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(4) (counted with multiplicity).

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 4, 8, 4, 2, 2, 4, 4, 2, 2, 8, 2, 2, 2, 4, 2, 4, 6, 16, 4, 4, 4, 4, 2, 2, 4, 8, 4, 4, 6, 4, 4, 2, 2, 16, 2, 2, 8, 4, 2, 2, 4, 8, 4, 2, 2, 8, 2, 6, 12, 32, 8, 4, 2, 8, 4, 4, 2, 8, 8, 2, 4, 4, 4, 4, 2, 16, 2, 4, 2, 8, 16, 6, 4, 8, 8, 4

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OFFSET

1,3

LINKS

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

FORMULA

Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(4,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.

EXAMPLE

Let GF(4) = GF(2)[w], where w^2 + w + 1 = 0. Factorizations of the n-th cyclotomic polynomial over GF(4) for n <= 10:

n = 1: x + 1;

n = 2: x + 1;

n = 3: (x + w)*(x + (w+1));

n = 4: (x + 1)^2;

n = 5: x^4 + x^3 + x^2 + x + 1;

n = 6: (x + w)*(x + (w+1));

n = 7: (x^3 + x + 1)*(x^3 + x^2 + 1);

n = 8: (x + 1)^4;

n = 9: (x^3 + w)*(x^3 + (w+1));

n = 10: x^4 + x^3 + x^2 + x + 1.

MATHEMATICA

a[n_] := EulerPhi[n] / MultiplicativeOrder[4, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jul 21 2024 *)

PROG

(PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(4, s))

CROSSREFS

Cf. A000010.

Row 3 of A327818.

Sequence in context: A076500 A344887 A060594 * A104361 A211449 A086876

Adjacent sequences: A327810 A327811 A327812 * A327814 A327815 A327816

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Sep 26 2019

STATUS

approved