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 1-4 of 4 results.

A132447 First primitive GF(2)[X] polynomial of degree n.

Original entry on oeis.org

3, 7, 11, 19, 37, 67, 131, 285, 529, 1033, 2053, 4179, 8219, 16427, 32771, 65581, 131081, 262183, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108935, 134217767, 268435465, 536870917, 1073741907, 2147483657
Offset: 1

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Examples

			a(5)=37, or 100101 in binary, representing the GF(2)[X] polynomial X^5+X^2+1, because it has degree 5 and is primitive, contrary to X^5, X^5+1, X^5+x^1, X^5+X^1+1 and X^5+X^2.

Links

Crossrefs

a(n) is the smallest member of A091250 at least 2^n. A132448(n) = a(n)-2^n, giving a more compact representation. Cf. A132449, similar, with restriction to at most 5 terms. Cf. A132451, similar, with restriction to exactly 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.

Programs

  • Maple
    f:= proc(n) local k,L,i,X;
   for k from 2^n+1 by 2 do
     L:= convert(k,base,2);
     if Primitive(add(L[i]*X^(i-1),i=1..n+1)) mod 2 then return k fi
   od
end proc:
map(f, [$1..40]); # Robert Israel, Nov 05 2023
  • Mathematica
    f[n_] := If[n == 1, 3, Module[{k, L, i, X}, For[k = 2^n+1, True, k = k+2, L = IntegerDigits[k, 2]; If[PrimitivePolynomialQ[Sum[L[[i]]*X^(i-1), {i, 1, n+1}], 2], Return[k]]]]];
Table[f[n], {n, 1, 40}] (* Jean-François Alcover, Mar 29 2024, after Robert Israel *)

A132449 First primitive GF(2)[X] polynomial of degree n with at most 5 terms.

Original entry on oeis.org

3, 7, 11, 19, 37, 67, 131, 285, 529, 1033, 2053, 4179, 8219, 16427, 32771, 65581, 131081, 262183, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108935, 134217767, 268435465, 536870917, 1073741907, 2147483657
Offset: 1

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of primitive GF(2)[X] polynomials of degree n with at most 5 terms. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software. The limitation to 5 terms occurs first for a(32), which is 4294967493 representing X^32+X^7+X^6+X^2+1, rather than 4294967471 representing X^32+X^7+X^5+X^3+X^2+X^1+1. Proof is needed that there exists a primitive GF(2)[X] polynomial P[X] of degree n and at most 5 terms for all positive n.

Examples

			a(5)=37, or 100101 in binary, representing the GF(2)[X] polynomial X^5+X^2+1, because it has degree 5 and no more than 5 terms and is primitive, contrary to X^5, X^5+1, X^5+X^1, X^5+X^1+1 and X^5+X^2.

Links

Crossrefs

Subset of A091250. A132450(n) = a(n)-2^n, giving a more compact representation. Cf. A132447, similar, with no restriction on number of terms. Cf. A132451, similar, with restriction to exactly 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.

A132452 First primitive GF(2)[X] polynomials of degree n with exactly 5 terms, X^n suppressed.

Original entry on oeis.org

15, 27, 15, 29, 27, 27, 23, 83, 27, 43, 23, 45, 15, 39, 39, 83, 39, 57, 43, 27, 15, 71, 39, 83, 23, 83, 15, 197, 83, 281, 387, 387, 83, 99, 147, 57, 15, 153, 89, 101, 27, 449, 51, 657, 113, 29, 75, 75, 71, 329, 71, 149, 45, 99, 149, 53, 39, 105, 51, 27, 27, 833, 39, 163, 101, 43, 43, 1545, 29
Offset: 5

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of GF(2)[X] polynomials P[X] of degree less than n and exactly 4 terms such that X^n+P[X] is primitive.
Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software.
Proof is needed that there exists a primitive GF(2)[X] polynomial P[X] of degree n and exactly 5 terms for all n>4.

Examples

			a(11)=23, or 10111 in binary, representing the GF(2)[X] polynomial X^4+X^2+X^1+1, because X^11+X^4+X^2+X^1+1 has exactly 5 terms and it is primitive, contrary to X^11+X^3+X^2+X^1+1.

Links

Crossrefs

For n>4, 2^n+a(n) belongs to A091250. A132451(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132448, similar, with no restriction on number of terms. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.

Extensions

Edited and extended by Max Alekseyev, Feb 06 2010

A132453 First primitive GF(2)[X] polynomial of degree n and minimal number of terms.

Original entry on oeis.org

3, 7, 11, 19, 37, 67, 131, 285, 529, 1033, 2053, 4179, 8219, 16427, 32771, 65581, 131081, 262273, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108935, 134217767, 268435465, 536870917, 1073741907, 2147483657
Offset: 1

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of primitive GF(2)[X] polynomials of degree n and minimal number of terms for such polynomials. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software.

Examples

			a(10)=1033, or 10000001001 in binary, representing the GF(2)[X] polynomial X^10+X^3+1, because this polynomial has degree 10, it has 3 terms and no degree 10 polynomial with less terms than that is primitive and it is primitive, contrary to X^10+X^1+1, X^10+X^2+1 and X^10+X^2+X^1.

Links

Crossrefs

Subset of A091250. A132454(n) encodes a(n) in a more compact representation. Cf. A132447, similar, with no restriction on number of terms. Cf. A132449, similar, with restriction to at most 5 terms. Cf. A132451, similar, with restriction to exactly 5 terms.
Showing 1-4 of 4 results.

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