cp's OEIS Frontend

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

%I A257688 #19 Jan 12 2025 14:56:09
%S A257688 1,2,3,5,7,11,13,17,19,23,25,29,31,37,41,43,47,53,55,59,61,67,71,73,
%T A257688 79,83,87,89,91,97,101,103,107,109,113,115,117,127,131,137,139,143,
%U A257688 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
%N A257688 After 1, all numbers that are either primes in Z or whose binary representation encodes a polynomial irreducible over GF(2).
%C A257688 "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).
%H A257688 Antti Karttunen, <a href="/A257688/b257688.txt">Table of n, a(n) for n = 1..10000</a>
%H A257688 <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials over GF(2)</a>
%o A257688 (PARI)
%o A257688 isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1,2)); \\ From _Charles R Greathouse IV_
%o A257688 isA257688(n) = ((1 == n) || isprime(n) || isA014580(n));
%o A257688 n = 0; i = 0; while(i < 10000, n++; if(isA257688(n), i++; write("b257688.txt", i, " ", n)));
%o A257688 (Scheme)
%o A257688 ;; With _Antti Karttunen_'s IntSeq-library.
%o A257688 (define A257688 (MATCHING-POS 1 1 (lambda (n) (or (= 1 n) (= 1 (A091225 n)) (= 1 (A010051 n))))))
%Y A257688 Union of A008578 and A014580.
%Y A257688 Complement of A091212 (Numbers that are composite in Z and reducible in ring GF(2)[X]).
%Y A257688 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.
%Y A257688 Differs from A186891 for the first time at n=22, where a(22) = 67, while A186891(22) = 65.
%Y A257688 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.
%K A257688 nonn
%O A257688 1,2
%A A257688 _Antti Karttunen_, May 07 2015