A327814 - OEIS

%I #10 Jul 21 2024 03:44:52

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

%H Amiram Eldar, <a href="/A327814/b327814.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 Factorizations of the n-th cyclotomic polynomial over GF(5) for n <= 10:

%e n = 1: x - 1;

%e n = 2: x + 1;

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

%e n = 4: (x + 2)*(x - 2);

%e n = 5: (x - 1)^4;

%e n = 6: x^2 - x + 1;

%e n = 7: x^6 + x^5 + x^4 + x^3 + x^2 + x + 1;

%e n = 8: (x^2 + 2)*(x^2 - 2);

%e n = 9: x^6 + x^3 + 1;

%e n = 10: (x + 1)^4.

%t a[n_] := EulerPhi[n] / MultiplicativeOrder[5, n / 5^IntegerExponent[n, 5]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *)

%o (PARI) a(n) = my(s=n/5^valuation(n, 5)); eulerphi(n)/znorder(Mod(5, s))

%Y Cf. A000010.

%Y Row 4 of A327818.

%K nonn,easy

%O 1,4

%A _Jianing Song_, Sep 26 2019