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.

A132448 First primitive polynomial over GF(2) of degree n, X^n suppressed.

Original entry on oeis.org

1, 3, 3, 3, 5, 3, 3, 29, 17, 9, 5, 83, 27, 43, 3, 45, 9, 39, 39, 9, 5, 3, 33, 27, 9, 71, 39, 9, 5, 83, 9, 175, 83, 231, 5, 119, 63, 99, 17, 57, 9, 63, 89, 101, 27, 303, 33, 183, 113, 29, 75, 9, 71, 125, 71, 149, 45, 99, 123, 3, 39, 105, 3, 27, 27, 365, 39, 163
Offset: 1

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Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of polynomials P[X] with coefficients in GF(2) such that X^n+P[X] is primitive. Applications include maximum-length linear feedback shift registers with efficient implementation in software.

Examples

			a(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.

Links

Crossrefs

2^n+a(n) is the smallest member of A091250 at least 2^n. A132447(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132452, similar, with restriction to exactly 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.

Programs

  • Mathematica
    i2px[i_]:=If[i>1,BitAnd[i,1]+i2px[BitShiftRight[i,1]]x,i ];s={1};For[n=2,n<69,++n,For[i=3,!PrimitivePolynomialQ[i2px[i]+x^n,2],i+=2];AppendTo[s,i]];s (* Francois R. Grieu, Jan 15 2021 *)

Extensions

More terms from Francois R. Grieu, Jan 12 2021

A132450 First primitive GF(2)[X] polynomials of degree n with at most 5 terms, X^n suppressed.

Original entry on oeis.org

1, 3, 3, 3, 5, 3, 3, 29, 17, 9, 5, 83, 27, 43, 3, 45, 9, 39, 39, 9, 5, 3, 33, 27, 9, 71, 39, 9, 5, 83, 9, 197, 83, 281, 5, 387, 83
Offset: 1

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of GF(2)[X] polynomials P[X] with at most 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. The limitation of the number of terms occurs first for a(32), which is 197 representing X^7+X^6+X^2+1, rather than 175 representing 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(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 has no more than 5 terms and X is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.

Links

Crossrefs

2^n+a(n) belongs to A091250. A132449(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132448 (similar, with no restriction on number of terms). Cf. A132452 (similar, with restriction to exactly 5 terms).

A132451 First primitive GF(2)[X] polynomials of degree n with exactly 5 terms.

Original entry on oeis.org

0, 0, 0, 0, 47, 91, 143, 285, 539, 1051, 2071, 4179, 8219, 16427, 32791, 65581, 131087, 262183, 524327, 1048659, 2097191, 4194361, 8388651, 16777243, 33554447, 67108935, 134217767, 268435539, 536870935, 1073741907, 2147483663
Offset: 1

Views

Author

Francois R. Grieu (f(AT)grieu.com), Aug 22 2007

Keywords

Comments

More precisely: minimum value for X=2 of primitive GF(2)[X] polynomials of degree n with exactly 5 terms, or 0 if no such polynomial exists. 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(6)=91, or 1011011 in binary, representing the GF(2)[X] polynomial X^6+X^4+X^3+X^1+1, because it has degree 6 and exactly 5 terms and is primitive, contrary to X^6+X^3+X^2+X^1+1 and X^6+X^4+X^2+X^1+1.

Links

Crossrefs

For n>4, a(n) belongs to A091250. A132452(n) = a(n)-2^n, giving a more compact representation. Cf. A132447, similar, with no restriction on number of terms. Cf. A132449, similar, with restriction to a most 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.

A132454 First primitive GF(2)[X] polynomials of degree n and minimal number of terms, expressed as -k for X^n+X^k+1, else with X^n suppressed.

Original entry on oeis.org

1, -1, -1, -1, -2, -1, -1, 29, -4, -3, -2, 83, 27, 43, -1, 45, -3, -7, 39, -3, -2, -1, -5, 27, -3, 71, 39, -3, -2, 83, -3, 197, -13, 281, -2, -11, 83
Offset: 1

Views

Author

Francois R. Grieu, Aug 22 2007

Keywords

Comments

More precisely: when there exists k, 0

Examples

			a(10)=-3, representing the GF(2)[X] polynomial X^10+X^3+1, because this degree 10 trinomial is primitive, contrary to X^10+X^1+1, X^10+X^2+1 and X^10+X^2+X^1.

Links

Crossrefs

Either of 2^n+2^(-a(n))+1 or 2^n+a(n) belongs to A091250. If there exists m such that n = A073726(m), then a(n) = -A074744(m); otherwise a(n) = A132450(n). A132453(n) gives the primitive polynomial corresponding to a(n). Cf. A132448, similar, with no restriction on number of terms. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132452, similar, with restriction to exactly 5 terms.
Showing 1-4 of 4 results.

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