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A327816 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(8) (counted with multiplicity).

%I A327816 #11 Jul 21 2024 03:44:17
%S A327816 1,1,1,2,1,1,6,4,3,1,1,2,3,6,2,8,2,3,3,2,6,1,2,4,1,3,3,12,1,2,6,16,2,
%T A327816 2,6,6,3,3,6,4,2,6,3,2,6,2,2,8,6,1,4,6,1,3,2,24,6,1,1,4,3,6,18,32,12,
%U A327816 2,3,4,2,6,2,12,24,3,2,6,6,6,6,8,3,2,1,12,8,3,2,4,8,6
%N A327816 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(8) (counted with multiplicity).
%H A327816 Amiram Eldar, <a href="/A327816/b327816.txt">Table of n, a(n) for n = 1..10000</a>
%F A327816 Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(8,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
%e A327816 Let GF(8) = GF(2)[y]/(y^3+y+1). Factorizations of the n-th cyclotomic polynomial over GF(8) for n <= 10:
%e A327816 n = 1: x + 1;
%e A327816 n = 2: x + 1;
%e A327816 n = 3: x^2 + x + 1;
%e A327816 n = 4: (x + 1)^2;
%e A327816 n = 5: x^4 + x^3 + x^2 + x + 1;
%e A327816 n = 6: x^2 + x + 1;
%e A327816 n = 7: (x + y)*(x + (y+1))*(x + y^2)*(x + (y^2+1))*(x + (y^2+y))*(x + (y^2+y+1));
%e A327816 n = 8: (x + 1)^4;
%e A327816 n = 9: (x^2 + y*x + 1)*(x^2 + (y+1)*x + 1)*(x^2 + y^2*x + 1);
%e A327816 n = 10: x^4 + x^3 + x^2 + x + 1.
%t A327816 a[n_] := EulerPhi[n] / MultiplicativeOrder[8, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *)
%o A327816 (PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(8, s))
%Y A327816 Cf. A000010.
%Y A327816 Row 6 of A327818.
%K A327816 nonn,easy
%O A327816 1,4
%A A327816 _Jianing Song_, Sep 26 2019