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A327817 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(9) (counted with multiplicity).

%I A327817 #10 Jul 21 2024 03:43:52
%S A327817 1,1,2,2,2,2,2,4,6,2,2,4,4,2,4,4,2,6,2,4,4,2,2,8,2,4,18,4,2,4,2,4,4,2,
%T A327817 4,12,4,2,8,8,10,4,2,4,12,2,2,8,2,2,4,8,2,18,4,8,4,2,2,8,12,2,12,4,8,
%U A327817 4,6,4,4,4,2,24,12,4,4,4,4,8,2,16,54,10,2,8,8,2,4,8,2,12
%N A327817 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(9) (counted with multiplicity).
%H A327817 Amiram Eldar, <a href="/A327817/b327817.txt">Table of n, a(n) for n = 1..10000</a>
%F A327817 Let n = 3^e*s, gcd(3,s) = 1, then a(n) = phi(n)/ord(9,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
%e A327817 Let GF(9) = GF(3)[i], where i^2 = -1. Factorizations of the n-th cyclotomic polynomial over GF(9) for n <= 10:
%e A327817 n = 1: x - 1;
%e A327817 n = 2: x + 1;
%e A327817 n = 3: (x - 1)^2;
%e A327817 n = 4: (x + i)*(x - i);
%e A327817 n = 5: (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
%e A327817 n = 6: (x + 1)^2;
%e A327817 n = 7: (x^3 + (-1+i)*x^2 + (1+i)*x - 1)*(x^3 + (-1-i)*x^2 + (1-i)*x - 1);
%e A327817 n = 8: (x + (1+i))*(x + (1-i))*(x + (-1+i))*(x + (-1-i));
%e A327817 n = 9: (x - 1)^6;
%e A327817 n = 10: (x^2 + (1+i)*x + 1)*(x^2 + (1-i)*x + 1).
%t A327817 a[n_] := EulerPhi[n] / MultiplicativeOrder[9, n / 3^IntegerExponent[n, 3]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *)
%o A327817 (PARI) a(n) = my(s=n/3^valuation(n, 3)); eulerphi(n)/znorder(Mod(9, s))
%Y A327817 Cf. A000010.
%Y A327817 Row 7 of A327818.
%K A327817 nonn,easy
%O A327817 1,3
%A A327817 _Jianing Song_, Sep 26 2019