cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A091242 Reducible polynomials over GF(2), coded in binary.

Original entry on oeis.org

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
Offset: 1

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Author

Antti Karttunen, Jan 03 2004

Keywords

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

Examples

			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].

Links

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.
Cf. A002808, A048720, A193231.

Programs

  • 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 *)

Extensions

Edited by M. F. Hasler, Aug 18 2014

A246156 Odd reducible polynomials over GF(2), coded in binary. (Polynomials with the constant term 1 that are reducible over GF(2)).

Original entry on oeis.org

5, 9, 15, 17, 21, 23, 27, 29, 33, 35, 39, 43, 45, 49, 51, 53, 57, 63, 65, 69, 71, 75, 77, 79, 81, 83, 85, 89, 93, 95, 99, 101, 105, 107, 111, 113, 119, 121, 123, 125, 127, 129, 133, 135, 139, 141, 147, 149, 151, 153, 155, 159, 161, 163, 165, 169, 173, 175, 177, 179, 181, 183, 187, 189, 195, 197, 199, 201
Offset: 1

Views

Author

Antti Karttunen, Aug 20 2014

Keywords

Comments

Self-inverse permutation A193231 maps each term of this sequence to some term of A246158 and vice versa.

Examples

			5, which is 101 in binary, encodes polynomial x^2 + 1, which factorizes as (x+1)(x+1) over GF(2), (5 = A048720(3,3)), thus it is reducible in that polynomial ring. Also, its constant term is 1, (not zero), thus 5 is a member of this sequence.

Links

Crossrefs

Intersection of A091242 and A005408 (odd numbers).
A246157 is a subsequence. Cf. also A048720, A193231, A246158.

Programs

  • Maple
    filter:= proc(n) local L,p,x;
    L:= convert(n,base,2);
    p:= add(L[i]*x^(i-1),i=1..nops(L));
    not (Irreduc(p) mod 2)
end proc:
select(filter,[seq(2*i+1,i=1..100)]); # Robert Israel, Aug 21 2014

A246158 Odious reducible polynomials over GF(2), coded in binary. (Polynomials with an odd number of nonzero terms that are reducible over GF(2)).

Original entry on oeis.org

4, 8, 14, 16, 21, 22, 26, 28, 32, 35, 38, 42, 44, 49, 50, 52, 56, 62, 64, 69, 70, 74, 76, 79, 81, 82, 84, 88, 93, 94, 98, 100, 104, 107, 110, 112, 118, 121, 122, 124, 127, 128, 133, 134, 138, 140, 146, 148, 151, 152, 155, 158, 161, 162, 164, 168, 173, 174, 176, 179, 181, 182, 186, 188, 194, 196, 199, 200
Offset: 1

Views

Author

Antti Karttunen, Aug 20 2014

Keywords

Comments

Self-inverse permutation A193231 maps each term of this sequence to some term of A246156 and vice versa.
Each term belongs into a distinct infinite cycle in permutations like A246161/A246162 and A246163/A246164 apart from 4, which is in a finite cycle (3 4) of A246161/A246162 and 4 and 8 which both are in the same (infinite) cycle of A246163/A246164.

Examples

			4, which is 100 in binary, encodes polynomial x^2, which factorizes as (x)(x) over GF(2), (4 = A048720(2,2)), thus it is reducible in that polynomial ring. It also has an odd number of nonzero terms present (equally: odd number of 1-bits in its code), in this case just one, thus 4 is a member of this sequence.

Links

Crossrefs

Intersection of A091242 and A000069 (odious numbers).
A238186 and A246157 are subsequences.
Cf. also A048720, A246156, A193231, A246161, A246162, A246163, A246164.
Showing 1-3 of 3 results.

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