A257688 After 1, all numbers that are either primes in Z or whose binary representation encodes a polynomial irreducible over GF(2).

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 211, 213, 223, 227, 229, 233, 239, 241, 247, 251, 253, 257, 263, 269

Offset: 1

Author

Antti Karttunen, May 07 2015

Keywords

• nonn

Comments

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to polynomials over GF(2)

Crossrefs

Union of A008578 and A014580.

Complement of A091212 (Numbers that are composite in Z and reducible in ring GF(2)[X]).

After the initial 1, a subsequence of A206074 (n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2), from which this differs for the first time at n=23, where a(23)=71, while A206074(22) = 69, the first term missing from here.

Differs from A186891 for the first time at n=22, where a(22) = 67, while A186891(22) = 65.

Differs from A257689 and A257691 for the first time at n=19, where a(19) = 55, while 55 is missing from both A257689 and A257691.

Programs

PARI

isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1,2)); \\ From Charles R Greathouse IV
isA257688(n) = ((1 == n) || isprime(n) || isA014580(n));
n = 0; i = 0; while(i < 10000, n++; if(isA257688(n), i++; write("b257688.txt", i, " ", n)));

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A257688 (MATCHING-POS 1 1 (lambda (n) (or (= 1 n) (= 1 (A091225 n)) (= 1 (A010051 n))))))