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 A091209 Primes whose binary representation encodes a polynomial reducible over GF(2). 25
 5, 17, 23, 29, 43, 53, 71, 79, 83, 89, 101, 107, 113, 127, 139, 149, 151, 163, 173, 179, 181, 197, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 383, 389, 401, 409, 421, 431, 439, 443, 449, 457, 461, 467, 479, 491, 503, 509, 521, 523 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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). Except for 3, all primes with even Hamming weight (A027699) are terms, see A238186 for the subsequence of primes with odd Hamming weight. [Joerg Arndt and Antti Karttunen, Feb 19 2014] LINKS Antti Karttunen, Table of n, a(n) for n = 1..71800 A. Karttunen, Scheme-program for computing this sequence. FORMULA a(n) = A000040(A091210(n)) = A091242(A091211(n)). Other identities. For all n >= 1: A235043(a(n)) = n. [A235043 works as a left inverse of this sequence.] MAPLE Primes:= select(isprime, [2, seq(2*i+1, i=1..1000)]): filter:= proc(n) local L, x;     L:= convert(n, base, 2);     Irreduc(add(L[i]*x^(i-1), i=1..nops(L))) mod 2; end proc: remove(filter, Primes); # Robert Israel, May 17 2015 MATHEMATICA Select[Prime[Range[2, 100]], !IrreduciblePolynomialQ[bb = IntegerDigits[#, 2]; Sum[bb[[k]] x^(k-1), {k, 1, Length[bb]}], Modulus -> 2]&] (* Jean-François Alcover, Feb 28 2016 *) PROG (PARI) forprime(p=2, 10^3, if( ! polisirreducible( Mod(1, 2)*Pol(binary(p)) ), print1(p, ", ") ) ); \\ Joerg Arndt, Feb 19 2014 CROSSREFS Intersection of A000040 and A091242. Disjoint union of A238186 and (A027699 \ {3}). Left inverse: A235043. Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)). Cf. also A235041-A235042, A234742. Sequence in context: A322985 A240031 A260427 * A307471 A226671 A226674 Adjacent sequences:  A091206 A091207 A091208 * A091210 A091211 A091212 KEYWORD nonn AUTHOR Antti Karttunen, Jan 03 2004 STATUS approved

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)