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A091248 Number of irreducible factors in the factorization of X^n + 1 over GF(2) (counted with multiplicity).

%I A091248 #23 Aug 20 2022 10:47:56
%S A091248 1,2,2,4,2,4,3,8,3,4,2,8,2,6,5,16,3,6,2,8,6,4,3,16,3,4,4,12,2,10,7,32,
%T A091248 5,6,6,12,2,4,5,16,3,12,4,8,8,6,3,32,5,6,8,8,2,8,5,24,5,4,2,20,2,14,
%U A091248 13,64,7,10,2,12,6,12,3,24,9,4,8,8,6,10,3,32,5,6,2,24,12,8,5,16,9,16
%N A091248 Number of irreducible factors in the factorization of X^n + 1 over GF(2) (counted with multiplicity).
%H A091248 Robert Israel, <a href="/A091248/b091248.txt">Table of n, a(n) for n = 1..10000</a>
%H A091248 Antti Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence</a>.
%H A091248 <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>.
%F A091248 a(n) = A091222(A000051(n)).
%F A091248 a(n) = Sum_{d|n} A318622(d). - _Robert Israel_, Aug 30 2018
%p A091248 h:= proc(n) option remember;  numtheory:-phi(n)/numtheory:-order(2,n/2^padic:-ordp(n,2)) end proc:
%p A091248 f:= n -> add(h(d),d=numtheory:-divisors(n)):
%p A091248 map(f, [$1..100]); # _Robert Israel_, Aug 30 2018
%t A091248 h[n_] := EulerPhi[n]/MultiplicativeOrder[2, n/2^IntegerExponent[n, 2]];
%t A091248 a[n_] := DivisorSum[n, h];
%t A091248 Array[a, 100] (* _Jean-François Alcover_, Aug 19 2022, after _Robert Israel_ *)
%Y A091248 Cf. A000374 (number of distinct irreducible factors of the same polynomials).
%Y A091248 Cf. A000051, A091222, A318622.
%K A091248 nonn
%O A091248 1,2
%A A091248 _Antti Karttunen_, Jan 03 2004