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

Author

Jianing Song, May 10 2021

Keywords

• nonn

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)).

Examples

			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.
		

Links

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

Crossrefs

Cf. A014580, A344141, A344142, A000120.

Programs

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.