A073726 Primitive irreducible trinomials: x^n + x^k + 1 is a primitive irreducible polynomial (mod 2) for some k with 0 < k < n.

2, 3, 4, 5, 6, 7, 9, 10, 11, 15, 17, 18, 20, 21, 22, 23, 25, 28, 29, 31, 33, 35, 36, 39, 41, 47, 49, 52, 55, 57, 58, 60, 63, 65, 68, 71, 73, 79, 81, 84, 87, 89, 93, 94, 95, 97, 98, 100, 103, 105, 106, 108, 111, 113, 118, 119, 121, 123, 124, 127, 129, 130, 132, 134, 135, 137, 140, 142, 145, 148, 150, 151, 153, 159, 161, 167, 169, 170, 172, 174, 175, 177, 178, 183, 185, 191, 193, 194, 198, 199, 201

Offset

1,1

Comments

Start is similar to A194125; first terms here but missing there are 140, 212, 236.

References

S. W. Golomb, "Shift Register Sequences", revised edition, reprinted by Aegean Park Press, 1982. See Tables V-1, V-2.

Links

Table of n, a(n) for n=1..91.

Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400.

Joerg Arndt, Complete list of primitive trinomials over GF(2) up to degree 400 [Cached copy, with permission]

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.8.

Index entries for sequences related to trinomials over GF(2)

Maple

A073726 := proc(n) local k, m: option remember: if(n=1)then return 2: else m:=procname(n-1)+1: while(true)do for k from 1 to m-1 do if Primitive(x^m+x^k+1) mod 2 then return m: fi: od: m:=m+1: od: fi: end:
seq(A073726(n), n=1..20); # Nathaniel Johnston, Apr 26 2011

Mathematica

okQ[n_] := AnyTrue[Range[n-1], PrimitivePolynomialQ[x^n + x^# + 1, 2]&];
Select[Range[201], okQ] (* Jean-François Alcover, Aug 19 2019 *)

Prog

(Magma) A073726 := function(n) for k := 1 to n-1 do if IsPrimitive(x^n+x^k+1) then return true; end if; end for; return false; end function; l := []; for n := 1 to 100 do if A073726(n) then l := Append(l, n); end if; end for; l;

Crossrefs

See A073571 for irreducible trinomials and A001153 for primitive Mersenne trinomials (and references). See A074744 for values of k.

Cf. A194125 (n such that x^n+(1+x)^w over GF(2) is primitive for some w).

Sequence in context: A271108 A179401 A117729 * A194125 A008839 A305925

Adjacent sequences: A073723 A073724 A073725 * A073727 A073728 A073729

Keyword

nonn,nice

Author

Richard P. Brent and Paul Zimmermann, Sep 05 2002

Extensions

a(49)-a(58) from Nathaniel Johnston, Apr 26 2011

Status

approved