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A305422 GF(2)[X] factorization prime shift towards smaller terms.

%I A305422 #15 Jan 20 2020 21:43:42
%S A305422 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 A305422 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 A305422 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 A305422 GF(2)[X] factorization prime shift towards smaller terms.
%C A305422 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 A305422 Antti Karttunen, <a href="/A305422/b305422.txt">Table of n, a(n) for n = 1..65537</a>
%H A305422 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>
%F A305422 For all n >= 1:
%F A305422 a(A305421(n)) = n.
%F A305422 a(A001317(n)) = A000079(n).
%F A305422 A007814(a(n)) = A268389(n).
%o A305422 (PARI)
%o A305422 A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
%o A305422 A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
%o A305422 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 A305422 Cf. A000079 (positions of ones), A014580, A091225, A268389, A305419, A305421, A305424 (odd bisection), A305425.
%Y A305422 Cf. also A064989, A300840.
%K A305422 nonn
%O A305422 1,3
%A A305422 _Antti Karttunen_, Jun 07 2018