login
GF(2)[X] factorization prime shift towards smaller terms.
10

%I #15 Jan 20 2020 21:43:42

%S 1,1,2,1,4,2,3,1,6,4,7,2,11,3,8,1,16,6,13,4,5,7,22,2,19,11,12,3,14,8,

%T 25,1,50,16,29,6,31,13,28,4,37,5,38,7,24,22,41,2,9,19,32,11,26,12,47,

%U 3,44,14,55,8,59,25,10,1,20,50,61,16,21,29,118,6,67,31,88,13,110,28,53,4,69,37,18,5,64,38,73,7,94,24,87,22,43,41,52,2,91

%N GF(2)[X] factorization prime shift towards smaller terms.

%C Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as f(i) x f(j) x ... x f(k), with 1 <= i <= j <= ... <= k, a(n) = f(i-1) x f(j-1) x ... x f(k-1), where f(0) = 1, and f(n) = A014580(n) for n >= 1.

%H Antti Karttunen, <a href="/A305422/b305422.txt">Table of n, a(n) for n = 1..65537</a>

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

%F For all n >= 1:

%F a(A305421(n)) = n.

%F a(A001317(n)) = A000079(n).

%F A007814(a(n)) = A268389(n).

%o (PARI)

%o A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));

%o A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));

%o A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };

%Y Cf. A000079 (positions of ones), A014580, A091225, A268389, A305419, A305421, A305424 (odd bisection), A305425.

%Y Cf. also A064989, A300840.

%K nonn

%O 1,3

%A _Antti Karttunen_, Jun 07 2018