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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
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).
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. also A235041-A235042.
Sequence in context: A108166 A080863 A339729 * A338009 A036305 A370351
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 03 2004
EXTENSIONS
Entry revised and name corrected by Antti Karttunen, May 17 2015
STATUS
approved