A235032 Numbers which are factored to the same set of primes in Z as to the binary codes of irreducible polynomials in GF(2)[X].

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 16, 19, 22, 24, 26, 28, 31, 32, 37, 38, 41, 44, 47, 48, 52, 56, 59, 61, 62, 64, 67, 73, 74, 76, 82, 88, 94, 96, 97, 103, 104, 109, 111, 112, 118, 122, 123, 124, 128, 131, 134, 137, 146, 148, 152, 157, 164, 167, 176, 188

Offset: 1

Author

Antti Karttunen, Jan 02 2014

Keywords

• nonn

Comments

This is a subsequence of the sequence which gives all such n that A001222(n) = A091222(n).

Examples

			2, 3 and 11 are included in this sequence, because they occur in A091206. That is, they are all primes, and encode irreducible polynomials in ring GF(2)[X] via their binary representations: For 2, '10' in binary, corresponds to polynomial x, and for 3, '11' in binary, corresponds to polynomial x+1, and for 11, '1011' in binary, corresponds to polynomial x^3+x+1, which are all irreducible in GF(2)[X].
4 is included in this sequence, because it factors as 2*2, but also because the corresponding GF(2)[X] polynomial x^2 factors as x*x (with the polynomial x encoded by the number 2).
5 is NOT included in this sequence, because, although it is prime, the corresponding polynomial (5 in binary is '101'): x^2 + 1 is not irreducible in GF(2)[X], but factors as (x+1)(x+1), i.e., we have 5 = A048720(3,3).
111 is included, as it is a product of two primes, 3*37, and these primes encode via their binary representations, '11' and '100101', two polynomials irreducible in GF(2)[X]: x+1 and x^5 + x^2 + 1, whose product, x^6 + x^5 + x^3 + x^2 + x + 1, is encoded by 111's binary representation, '1101111'.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..3048
• Index entries for sequences operating on (or containing) GF(2)[X]-polynomials

Crossrefs

Complement: A235033. Intersection of A235034 & A235035. Union of A091206 & A235036. Subsequence of A235045.

A235036 and A235039 give composite and odd composite (after 1) terms occurring in this sequence.

Gives the positions of zeros in A236380, i.e. such n that A234741(n) = A234742(n).

Cf. also A048720.