A091212 Composite numbers whose binary representation encodes a polynomial reducible over GF(2).

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99

Offset: 1

Author

Antti Karttunen, Jan 03 2004

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).

From Reinhard Zumkeller, Jul 05-12 2011, values for maximum n corrected by Antti Karttunen, May 18 2015: (Start)

a(n) = A192506(n) for n <= 36.

a(n) = A175526(n) for n <= 36.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 1..55429
• A. Karttunen, Scheme-program for computing beginning of this sequence.
• Index entries for sequences related to binary encoded polynomials over GF(2)

Crossrefs

Intersection of A002808 and A091242.

Cf. A257688 (complement, either 1, irreducible in GF(2)[X] or prime), A091206 (prime and irreducible), A091209 (prime and reducible), A091214 (nonprime and irreducible).

Cf. A091213, A236861, A235036 (a subsequence, apart from 1).

Differs from both A175526 and A192506 for the first time at n=37, where a(37) = 56, while A175526(37) = A192506(37) = 55, a term missing from here (as 55 encodes a polynomial which is irreducible in GF(2)[X]).

Differs from its subsequence A205783(n+1) for the first time at n=47, where a(47) = 69, while 69 is missing from A205783.

Programs

PARI

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091212(n) = ((n > 1) && !isprime(n) && !isA014580(n));
n = 0; i = 0; while(n < 2^16, n++; if(isA091212(n), i++; write("b091212.txt", i, " ", n)));

Formula

a(n) = A091242(A091213(n)).