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A327814 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(5) (counted with multiplicity).

%I A327814 #10 Jul 21 2024 03:44:52
%S A327814 1,1,1,2,4,1,1,2,1,4,2,2,3,1,4,2,1,1,2,8,2,2,1,4,20,3,1,2,2,4,10,2,2,
%T A327814 1,4,2,1,2,6,8,2,2,1,4,4,1,1,4,1,20,2,6,1,1,8,4,2,2,2,8,2,10,6,2,12,2,
%U A327814 3,2,2,4,14,4,1,1,20,4,2,6,2,8,1,2,1,4,4,1,4,4,2,4
%N A327814 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(5) (counted with multiplicity).
%H A327814 Amiram Eldar, <a href="/A327814/b327814.txt">Table of n, a(n) for n = 1..10000</a>
%F A327814 Let n = 5^e*s, gcd(5,s) = 1, then a(n) = phi(n)/ord(5,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information.
%e A327814 Factorizations of the n-th cyclotomic polynomial over GF(5) for n <= 10:
%e A327814 n = 1: x - 1;
%e A327814 n = 2: x + 1;
%e A327814 n = 3: x^2 + x + 1;
%e A327814 n = 4: (x + 2)*(x - 2);
%e A327814 n = 5: (x - 1)^4;
%e A327814 n = 6: x^2 - x + 1;
%e A327814 n = 7: x^6 + x^5 + x^4 + x^3 + x^2 + x + 1;
%e A327814 n = 8: (x^2 + 2)*(x^2 - 2);
%e A327814 n = 9: x^6 + x^3 + 1;
%e A327814 n = 10: (x + 1)^4.
%t A327814 a[n_] := EulerPhi[n] / MultiplicativeOrder[5, n / 5^IntegerExponent[n, 5]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *)
%o A327814 (PARI) a(n) = my(s=n/5^valuation(n, 5)); eulerphi(n)/znorder(Mod(5, s))
%Y A327814 Cf. A000010.
%Y A327814 Row 4 of A327818.
%K A327814 nonn,easy
%O A327814 1,4
%A A327814 _Jianing Song_, Sep 26 2019