

A091214


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


21



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
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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, Schemeprogram 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 bfile 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 A235041A235042.
Sequence in context: A176275 A108166 A080863 * A036305 A257708 A266817
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



