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Difference in number of prime factors (when counted with multiplicity) between GF(2)[X] (carryless binary) and ordinary factorization: a(n) = A091222(n) - A001222(n).
3

%I #13 Jul 02 2018 07:03:14

%S 0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,3,0,0,1,0,0,1,0,-1,0,0,0,1,1,0,0,0,3,

%T 0,0,0,0,1,1,0,0,1,0,1,1,0,0,0,-1,3,0,1,0,-1,0,1,1,0,1,0,0,0,0,2,0,0,

%U 3,0,0,1,0,0,0,1,0,0,1,1,1,-2,0,2,0,4,1,-1,0,1,1,-1,1,0,0,1,0,0,0,0,-1,2,3,0,0,1

%N Difference in number of prime factors (when counted with multiplicity) between GF(2)[X] (carryless binary) and ordinary factorization: a(n) = A091222(n) - A001222(n).

%H Antti Karttunen, <a href="/A305802/b305802.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ge#GF2X">Index entries for sequences operating on polynomials in ring GF(2)[X]</a>

%F a(n) = A091222(n) - A001222(n).

%F For all n, a(A091206(n)) = 0. [Note that zeros occur in other positions as well.]

%o (PARI)

%o A091222(n) = vecsum(factor(Pol(binary(n))*Mod(1, 2))[, 2]);

%o A305802(n) = (A091222(n) - bigomega(n));

%Y Cf. A001222, A091222, A091206, A305816.

%Y Cf. also A305789.

%K sign

%O 1,17

%A _Antti Karttunen_, Jun 10 2018