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

25, 55, 87, 91, 115, 117, 143, 145, 171, 185, 203, 213, 247, 253, 285, 299, 301, 319, 333, 351, 355, 357, 361, 369, 375, 391, 395, 415, 425, 445, 451, 471, 477, 501, 505, 515, 529, 535, 539, 545, 623, 637, 665, 675, 687, 695, 721, 731, 789, 799, 803, 817

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).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..48410; all terms up to binary length 20
• 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 A014580.

Subsequence of A235033, A236834 and A236838.

Left inverse: A235044.

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)).

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

Cf. also A235041-A235042.

Programs

Mathematica

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 *)

PARI

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091214(n) = (!isprime(n) && isA014580(n));
n = 0; i = 0; while(n < 2^20, n++; if(isA091214(n), i++; write("b091214.txt", i, " ", n)));
\\ The b-file was computed with this program. Antti Karttunen, May 17 2015

Formula

Other identities. For all n >= 1:

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

a(n) = A014580(A091215(n)). - Antti Karttunen, May 17 2015

Extensions

Entry revised and name corrected by Antti Karttunen, May 17 2015