This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327813 #11 Jul 21 2024 03:45:03 %S A327813 1,1,2,2,2,2,2,4,2,2,2,4,2,2,4,8,4,2,2,4,4,2,2,8,2,2,2,4,2,4,6,16,4,4, %T A327813 4,4,2,2,4,8,4,4,6,4,4,2,2,16,2,2,8,4,2,2,4,8,4,2,2,8,2,6,12,32,8,4,2, %U A327813 8,4,4,2,8,8,2,4,4,4,4,2,16,2,4,2,8,16,6,4,8,8,4 %N A327813 Number of irreducible factors in the factorization of the n-th cyclotomic polynomial over GF(4) (counted with multiplicity). %H A327813 Amiram Eldar, <a href="/A327813/b327813.txt">Table of n, a(n) for n = 1..10000</a> %F A327813 Let n = 2^e*s, gcd(2,s) = 1, then a(n) = phi(n)/ord(4,s), where phi = A000010, ord(k,s) is the multiplicative order of k modulo s. See A327818 for further information. %e A327813 Let GF(4) = GF(2)[w], where w^2 + w + 1 = 0. Factorizations of the n-th cyclotomic polynomial over GF(4) for n <= 10: %e A327813 n = 1: x + 1; %e A327813 n = 2: x + 1; %e A327813 n = 3: (x + w)*(x + (w+1)); %e A327813 n = 4: (x + 1)^2; %e A327813 n = 5: x^4 + x^3 + x^2 + x + 1; %e A327813 n = 6: (x + w)*(x + (w+1)); %e A327813 n = 7: (x^3 + x + 1)*(x^3 + x^2 + 1); %e A327813 n = 8: (x + 1)^4; %e A327813 n = 9: (x^3 + w)*(x^3 + (w+1)); %e A327813 n = 10: x^4 + x^3 + x^2 + x + 1. %t A327813 a[n_] := EulerPhi[n] / MultiplicativeOrder[4, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2024 *) %o A327813 (PARI) a(n) = my(s=n/2^valuation(n, 2)); eulerphi(n)/znorder(Mod(4, s)) %Y A327813 Cf. A000010. %Y A327813 Row 3 of A327818. %K A327813 nonn,easy %O A327813 1,3 %A A327813 _Jianing Song_, Sep 26 2019