A344143 Indices k such that A344141(k) and A344142(k) are not equal.

33, 34, 36, 37, 42, 49, 54, 55, 58, 59, 62, 65, 68, 71, 72, 73, 74, 76, 78, 79, 80, 82, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 98, 100, 102, 103, 106, 107, 108, 110, 111, 113, 115, 118, 121, 124, 125, 126, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141

Offset

1,1

Comments

A344141 and A344142 are two different methods of finding the "first irreducible GF(2)[X] polynomial of degree k". Sequence gives k such that this two methods disagree.

Obviously, k is a term if and only if A000120(A344141(k)) != A000120(A344142(k)).

Links

Jianing Song, Table of n, a(n) for n = 1..725

Example

33 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 33 is x^33 + x^6 + x^3 + x + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^33 + x^10 + 1.
37 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 37 is x^37 + x^5 + x^4 + x^3 + x^2 + x + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^37 + x^6 + x^4 + x + 1.
54 is a term, since lexicographically the first irreducible GF(2)[X] polynomial of degree 54 is x^54 + x^6 + x^5 + x^4 + x^3 + x^2 + 1, while lexicographically the first irreducible GF(2)[X] polynomial with the lowest possible number of terms is x^54 + x^9 + 1.

Prog

(PARI) isA344143(n) = my(k=A344142(n)-1); while(k>=2^n, if(polisirreducible(Mod(Pol(binary(k)), 2)), return(1), k--)); 0 \\ See A344142 for its program, assuming that an irreducible polynomial of degree n with at most 5 terms exists for every n.

Crossrefs

Cf. A014580, A344141, A344142, A000120.

Sequence in context: A165854 A112817 A115394 * A335072 A260678 A329658

Adjacent sequences: A344140 A344141 A344142 * A344144 A344145 A344146

Keyword

nonn

Author

Jianing Song, May 10 2021

Status

approved