

A327813


Number of irreducible factors in the factorization of the nth 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
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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 nth 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



