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A327815 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(7) (counted with multiplicity).

%I A327815 #10 Jul 21 2024 03:44:36
%S A327815 1,1,2,1,1,2,6,2,2,1,1,2,1,6,2,4,1,2,6,2,12,1,1,4,5,1,2,6,4,2,2,4,2,1,
%T A327815 6,2,4,6,2,4,1,12,7,2,2,1,2,8,42,5,2,2,2,2,2,12,12,4,2,4,1,2,12,4,4,2,
%U A327815 1,2,2,6,1,4,3,4,10,6,6,2,1,8,2,1,2,12,4,7,8,4,1,2
%N A327815 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(7) (counted with multiplicity).
%H A327815 Amiram Eldar, <a href="/A327815/b327815.txt">Table of n, a(n) for n = 1..10000</a>
%F A327815 Let n = 7^e*s, gcd(7,s) = 1, then a(n) = phi(n)/ord(7,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
%e A327815 Factorizations of the n-th cyclotomic polynomial over GF(7) for n <= 10:
%e A327815 n = 1: x - 1;
%e A327815 n = 2: x + 1;
%e A327815 n = 3: (x + 3)*(x - 2);
%e A327815 n = 4: x^2 + 1;
%e A327815 n = 5: x^4 + x^3 + x^2 + x + 1;
%e A327815 n = 6: (x + 2)*(x - 3);
%e A327815 n = 7: (x - 1)^6;
%e A327815 n = 8: (x^2 + 3x + 1)*(x^2 - 3x + 1);
%e A327815 n = 9: (x^3 + 3)*(x^3 - 2);
%e A327815 n = 10: x^4 - x^3 + x^2 - x^2 + 1.
%t A327815 a[n_] := EulerPhi[n] / MultiplicativeOrder[7, n / 7^IntegerExponent[n, 7]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *)
%o A327815 (PARI) a(n) = my(s=n/7^valuation(n, 7)); eulerphi(n)/znorder(Mod(7, s))
%Y A327815 Cf. A000010.
%Y A327815 Row 5 of A327818.
%K A327815 nonn,easy
%O A327815 1,3
%A A327815 _Jianing Song_, Sep 26 2019