A327813 - OEIS

%I #11 Jul 21 2024 03:45:03

%S 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,

%T 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,

%U 8,4,4,2,8,8,2,4,4,4,4,2,16,2,4,2,8,16,6,4,8,8,4

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

%H Amiram Eldar, <a href="/A327813/b327813.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%e 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:

%e n = 1: x + 1;

%e n = 2: x + 1;

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

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

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

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

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

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

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

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

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

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

%Y Cf. A000010.

%Y Row 3 of A327818.

%K nonn,easy

%O 1,3

%A _Jianing Song_, Sep 26 2019