cp's OEIS Frontend

A091214 Composite numbers whose binary representation encodes a polynomial irreducible over GF(2).

%I A091214 #27 May 18 2015 10:37:11
%S A091214 25,55,87,91,115,117,143,145,171,185,203,213,247,253,285,299,301,319,
%T A091214 333,351,355,357,361,369,375,391,395,415,425,445,451,471,477,501,505,
%U A091214 515,529,535,539,545,623,637,665,675,687,695,721,731,789,799,803,817
%N A091214 Composite numbers whose binary representation encodes a polynomial irreducible over GF(2).
%C A091214 "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 A091214 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 A091214 A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing beginning of this sequence.</a>
%H A091214 <a href="/index/Ge#GF2X">Index entries for sequences related to binary encoded polynomials over GF(2)</a>
%F A091214 Other identities. For all n >= 1:
%F A091214 A235044(a(n)) = n. [A235044 works as a left inverse of this sequence.]
%F A091214 a(n) = A014580(A091215(n)). - _Antti Karttunen_, May 17 2015
%t A091214 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 A091214 (PARI)
%o A091214 isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_
%o A091214 isA091214(n) = (!isprime(n) && isA014580(n));
%o A091214 n = 0; i = 0; while(n < 2^20, n++; if(isA091214(n), i++; write("b091214.txt", i, " ", n)));
%o A091214 \\ The b-file was computed with this program. _Antti Karttunen_, May 17 2015
%Y A091214 Intersection of A002808 and A014580.
%Y A091214 Subsequence of A235033, A236834 and A236838.
%Y A091214 Left inverse: A235044.
%Y A091214 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 A091214 Cf. A091215, A235027, A235046, A236841, A236845, A236850, A236851, A236861.
%Y A091214 Cf. also A235041-A235042.
%K A091214 nonn
%O A091214 1,1
%A A091214 _Antti Karttunen_, Jan 03 2004
%E A091214 Entry revised and name corrected by _Antti Karttunen_, May 17 2015