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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

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

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.

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: A083533 A076500 A060594 * A104361 A211449 A086876

Adjacent sequences:  A327810 A327811 A327812 * A327814 A327815 A327816

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Sep 26 2019

STATUS

approved

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Last modified June 4 01:32 EDT 2020. Contains 334809 sequences. (Running on oeis4.)