A236838 -id:A236838 - cp's OEIS frontend

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

Antti Karttunen, Jan 03 2004

nonn

"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).

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.

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 *)

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

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

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

A236834 Numbers that do not occur as results of "downward" remultiplication (N -> GF(2)[X]) of any number; numbers not present in A234741.

Original entry on oeis.org

25, 50, 55, 87, 91, 100, 110, 115, 117, 133, 143, 145, 159, 171, 174, 182, 185, 200, 203, 213, 220, 230, 234, 237, 247, 249, 253, 266, 267, 279, 285, 286, 290, 299, 301, 318, 319, 321, 333, 339, 342, 345, 348, 351, 355, 357, 361, 364, 369, 370, 375, 385, 391, 395, 400

Offset: 1

Antti Karttunen, Jan 31 2014

nonn

Numbers that do not occur in A234741 (A236841).
This is a subsequence of A236838, thus all terms are GF(2)[X]-multiples of some of the terms of A091214. (Cf. also A236844).
a(5)=91 is the first term that does not occur in A236849. On the other hand, A236849(4)=75, is the first term in the latter which does not occur here.

Antti Karttunen, Table of n, a(n) for n = 1..2525

Complement: A236841. This sequence is a setwise difference of A236838 and A236839.
A091214 is a subsequence.
Positions of zeros in A236833, A236836, A236837 and A236861
Cf. also A236844, A234741, A236835.

A236848 Numbers that have at least one prime divisor encoding a reducible polynomial in ring GF(2)[X]; multiples of terms of A091209.

Original entry on oeis.org

5, 10, 15, 17, 20, 23, 25, 29, 30, 34, 35, 40, 43, 45, 46, 50, 51, 53, 55, 58, 60, 65, 68, 69, 70, 71, 75, 79, 80, 83, 85, 86, 87, 89, 90, 92, 95, 100, 101, 102, 105, 106, 107, 110, 113, 115, 116, 119, 120, 125, 127, 129, 130, 135, 136, 138, 139, 140, 142, 145, 149, 150

Offset: 1

Antti Karttunen, Jan 31 2014

nonn

Numbers that are divisible by at least one prime whose binary representation encodes a polynomial which is reducible in polynomial ring GF(2)[X].

Antti Karttunen, Table of n, a(n) for n = 1..12311

Disjoint union of A236844 and A236849.
Complement: A236860.
Cf. A091209, A236838.

A236839 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number and have at least one irreducible polynomial factor encoded by a composite number.

Original entry on oeis.org

43, 79, 86, 89, 125, 135, 149, 158, 172, 178, 181, 209, 227, 235, 250, 263, 270, 281, 293, 298, 311, 316, 317, 331, 344, 349, 356, 362, 371, 383, 393, 399, 401, 418, 421, 443, 447, 454, 470, 479, 493, 500, 509, 519, 521, 523, 526, 540, 547, 562, 567, 577, 586, 596, 609

Offset: 1

Antti Karttunen, Feb 02 2014

nonn

Terms of A236841 that are found in A236838.

			43, which is A234741(43), and in binary '101011', encodes polynomial x^5 + x^3 + x + 1, which factors as (x + 1)(x^4 + x^3 + 1), i.e. 43 = A048720(3,25), and the latter factor of these, encoded by 25, is a composite in N, thus 43 is a term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..1605

Setwise difference A236838 \ A236834, and also A236841 \ A236850.

Cf. A236838, A236841, A236849.

cp's OEIS Frontend

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

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A236834 Numbers that do not occur as results of "downward" remultiplication (N -> GF(2)[X]) of any number; numbers not present in A234741.

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A236848 Numbers that have at least one prime divisor encoding a reducible polynomial in ring GF(2)[X]; multiples of terms of A091209.

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A236839 Numbers that occur as results of downward remultiplication (N -> GF(2)[X]) of some number and have at least one irreducible polynomial factor encoded by a composite number.

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