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A091242 Reducible polynomials over GF(2), coded in binary. 49
4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
"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
The reducible polynomials in GF(2)[X] are the analog to the composite numbers A002808 in the integers.
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
LINKS
EXAMPLE
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].
MAPLE
filter:= proc(n) local L;
L:= convert(n, base, 2);
not Irreduc(add(L[i]*x^(i-1), i=1..nops(L))) mod 2
end proc:
select(filter, [$2..200]); # Robert Israel, Aug 30 2018
MATHEMATICA
okQ[n_] := Module[{x, id = IntegerDigits[n, 2] // Reverse}, !IrreduciblePolynomialQ[id.x^Range[0, Length[id]-1], Modulus -> 2]];
Select[Range[2, 200], okQ] (* Jean-François Alcover, Jan 04 2022 *)
CROSSREFS
Inverse: A091246. Almost complement of A014580. Union of A091209 & A091212. First differences: A091243. Characteristic function: A091247. In binary format: A091254.
Number of degree-n reducible polynomials: A058766.
Subsequences: A001969\{0,3}, A005843\{0,2}, A246156, A246157, A246158, A316970.
Sequence in context: A284902 A023851 A285279 * A354834 A089585 A089253
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 03 2004
EXTENSIONS
Edited by M. F. Hasler, Aug 18 2014
STATUS
approved

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Last modified March 29 03:51 EDT 2024. Contains 371264 sequences. (Running on oeis4.)