A091214 - OEIS

%I #27 May 18 2015 10:37:11

%S 25,55,87,91,115,117,143,145,171,185,203,213,247,253,285,299,301,319,

%T 333,351,355,357,361,369,375,391,395,415,425,445,451,471,477,501,505,

%U 515,529,535,539,545,623,637,665,675,687,695,721,731,789,799,803,817

%N Composite numbers 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="/A091214/b091214.txt">Table of n, a(n) for n = 1..48410; all terms up to binary length 20</a>

%H A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing beginning of this sequence.</a>

%H <a href="/index/Ge#GF2X">Index entries for sequences related to binary encoded polynomials over GF(2)</a>

%F Other identities. For all n >= 1:

%F A235044(a(n)) = n. [A235044 works as a left inverse of this sequence.]

%F a(n) = A014580(A091215(n)). - _Antti Karttunen_, May 17 2015

%t fQ[n_] := Block[{ply = Plus @@ (Reverse@ IntegerDigits[n, 2] x^Range[0, Floor@ Log2@ n])}, ply == Factor[ply, Modulus -> 2] && n != 2^Floor@ Log2@ n && ! PrimeQ@ n]; Select[ Range@ 840, fQ] (* _Robert G. Wilson v_, Aug 12 2011 *)

%o (PARI)

%o isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_

%o isA091214(n) = (!isprime(n) && isA014580(n));

%o n = 0; i = 0; while(n < 2^20, n++; if(isA091214(n), i++; write("b091214.txt", i, " ", n)));

%o \\ The b-file was computed with this program. _Antti Karttunen_, May 17 2015

%Y Intersection of A002808 and A014580.

%Y Subsequence of A235033, A236834 and A236838.

%Y Left inverse: A235044.

%Y Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091209 (Primes that encode a polynomial reducible over GF(2)), A091212 (Composite, and reducible over GF(2)).

%Y Cf. A091215, A235027, A235046, A236841, A236845, A236850, A236851, A236861.

%Y Cf. also A235041-A235042.

%K nonn

%O 1,1

%A _Antti Karttunen_, Jan 03 2004

%E Entry revised and name corrected by _Antti Karttunen_, May 17 2015