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%I #15 May 14 2015 15:58:21
%S 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 79,83,87,89,91,97,101,103,107,109,113,115,117,127,131,137,139,143,
%U 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 After 1, all numbers that are either primes in Z or whose binary representation encodes a polynomial irreducible over GF(2).
%C "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 Antti Karttunen, <a href="/A257688/b257688.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials over GF(2)</a>
%o (PARI)
%o isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1,2)); \\ From _Charles R Greathouse IV_
%o isA257688(n) = ((1 == n) || isprime(n) || isA014580(n));
%o n = 0; i = 0; while(i < 10000, n++; if(isA257688(n), i++; write("b257688.txt", i, " ", n)));
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A257688 (MATCHING-POS 1 1 (lambda (n) (or (= 1 n) (= 1 (A091225 n)) (= 1 (A010051 n))))))
%Y Union of A008578 and A014580.
%Y Complement of A091212 (Numbers that are composite in Z and reducible in ring GF(2)[X]).
%Y 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 Differs from A186891 for the first time at n=22, where a(22) = 67, while A186891(22) = 65.
%Y 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 nonn
%O 1,2
%A _Antti Karttunen_, May 07 2015
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