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

1,1

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).

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)

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

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 *)

Prog

(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

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.

Sequence in context: A108166 A080863 A339729 * A338009 A036305 A370351

Adjacent sequences: A091211 A091212 A091213 * A091215 A091216 A091217

Keyword

nonn

Author

Antti Karttunen, Jan 03 2004

Extensions

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

Status

approved