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%I #32 Jan 04 2022 04:08:56
%S 4,5,6,8,9,10,12,14,15,16,17,18,20,21,22,23,24,26,27,28,29,30,32,33,
%T 34,35,36,38,39,40,42,43,44,45,46,48,49,50,51,52,53,54,56,57,58,60,62,
%U 63,64,65,66,68,69,70,71,72,74,75,76,77,78,79,80,81,82,83,84,85,86,88
%N Reducible polynomials over GF(2), coded in binary.
%C "Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1). - _M. F. Hasler_, Aug 18 2014
%C The reducible polynomials in GF(2)[X] are the analog to the composite numbers A002808 in the integers.
%C It follows that the sequence is closed under application of A048720(.,.), which effects multiplication of the coded polynomials. It is also closed under application of blue code, A193231. The majority of the terms are coded multiples of X^1 (represented by 2) and/or X^1+1 (represented by 3): see A005843 and A001969 respectively. A246157 lists the other terms. - _Peter Munn_, Apr 20 2021
%H Robert Israel, <a href="/A091242/b091242.txt">Table of n, a(n) for n = 1..10000</a>
%H Antti Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%e For example, 5 = 101 in binary encodes the polynomial x^2+1 which is factored as (x+1)^2 in the polynomial ring GF(2)[X].
%p filter:= proc(n) local L;
%p L:= convert(n,base,2);
%p not Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2
%p end proc:
%p select(filter, [$2..200]); # _Robert Israel_, Aug 30 2018
%t okQ[n_] := Module[{x, id = IntegerDigits[n, 2] // Reverse}, !IrreduciblePolynomialQ[id.x^Range[0, Length[id]-1], Modulus -> 2]];
%t Select[Range[2, 200], okQ] (* _Jean-François Alcover_, Jan 04 2022 *)
%Y Inverse: A091246. Almost complement of A014580. Union of A091209 & A091212. First differences: A091243. Characteristic function: A091247. In binary format: A091254.
%Y Number of degree-n reducible polynomials: A058766.
%Y Subsequences: A001969\{0,3}, A005843\{0,2}, A246156, A246157, A246158, A316970.
%Y Cf. A002808, A048720, A193231.
%K nonn
%O 1,1
%A _Antti Karttunen_, Jan 03 2004
%E Edited by _M. F. Hasler_, Aug 18 2014
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