A091209 -id:A091209 - cp's OEIS frontend

A235043 Partial sums of the characteristic function of A091209.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Note that this also works as an inverse function of A091209 in a sense that a(A091209(n)) = n for all n>=1.

Antti Karttunen, Table of n, a(n) for n = 0..10007

Used to compute A235041. Cf. A010051, A091247, A235044.

(define (A235043 n) (if (zero? n) n (+ (A235043 (- n 1)) (+ (* (A010051 n) (A091247 n))))))

A260716 a(n) = number of iterations of A234742 needed when starting from A091209(n) before a fixed point is reached.

Original entry on oeis.org

6, 4, 55, 141, 2, 2, 4, 5, 3, 4, 3, 14, 2, 1, 4, 3, 1, 18, 6, 3, 17, 36, 1, 10, 13, 1, 10, 2, 2, 86, 27, 7, 4, 50, 1, 4, 6, 4, 3, 13, 7, 3, 1, 207, 2, 7, 10, 10, 128, 7, 2, 4, 2, 9, 20, 2, 15, 24, 3, 10, 64, 7, 4, 4, 1, 4, 15, 8, 4, 1, 45, 3, 2, 1, 1, 2, 6, 28, 1, 2, 11, 1, 3, 14, 13, 3, 11, 12, 4, 28, 3, 7, 55, 40, 9, 4, 51, 5, 2, 6, 1, 2, 1, 15, 1

Offset: 1

Antti Karttunen, Aug 04 2015

nonn

It is not known whether the sequence is well-defined for all values. For example, does a(144) have a finite value? Cf. the sequence A260441, starting iteration from 1361 = A091209(144).

Antti Karttunen, Table of n, a(n) for n = 1..143

Cf. A091209, A234742, A260712, A260713, A260441.

allocatemem((2^29));
v091209 = [5, 17, 23, 29, 43, 53, 71, 79, 83, 89, 101, 107, 113, 127, 139, 149, 151, 163, 173, 179, 181, 197, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 383, 389, 401, 409, 421, 431, 439, 443, 449, 457, 461, 467, 479, 491, 503, 509, 521, 523, 541, 547, 569, 571, 577, 593, 599, 619, 641, 643, 653, 659, 673, 683, 691, 709, 727, 733, 739, 743, 751, 773, 797, 809, 811, 821, 823, 829, 839, 853, 857, 863, 881, 887, 907, 919, 937, 941, 947, 977, 983, 991, 997, 1009, 1013, 1021, 1031, 1049, 1061, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1129, 1151, 1171, 1181, 1187, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1283, 1289, 1297, 1301, 1303, 1307, 1319, 1321, 1327];
A091209(n) = v091209[n];
A234742(n) = factorback(subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2)); \\ After M. F. Hasler's Feb 18 2014 code.
A260712(n) = {my(prev=-1,i=-1); until((n==prev), prev = n; n = A234742(n); i++); return(i); }
A260716(n) = A260712(A091209(n));
for(n=1, 143, write("b260716.txt", n, " ", A260716(n)));

(define (A260716 n) (A260712 (A091209 n)))

a(n) = A260712(A091209(n)).

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.

A091242 Reducible polynomials over GF(2), coded in binary.

Original entry on oeis.org

4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

"Coded in binary" 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 a(k)=0 or 1). - M. F. Hasler, Aug 18 2014
The reducible polynomials in GF(2)[X] are the analog to the composite numbers A002808 in the integers.
It follows that the sequence is closed under application of A048720(.,.), which effects multiplication of the coded polynomials. It is also closed under application of blue code, A193231. The majority of the terms are coded multiples of X^1 (represented by 2) and/or X^1+1 (represented by 3): see A005843 and A001969 respectively. A246157 lists the other terms. - Peter Munn, Apr 20 2021

			For example, 5 = 101 in binary encodes the polynomial x^2+1 which is factored as (x+1)^2 in the polynomial ring GF(2)[X].

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials

Inverse: A091246. Almost complement of A014580. Union of A091209 & A091212. First differences: A091243. Characteristic function: A091247. In binary format: A091254.
Number of degree-n reducible polynomials: A058766.
Subsequences: A001969\{0,3}, A005843\{0,2}, A246156, A246157, A246158, A316970.
Cf. A002808, A048720, A193231.

filter:= proc(n) local L;
  L:= convert(n,base,2);
  not Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2
end proc:
select(filter, [$2..200]); # Robert Israel, Aug 30 2018

okQ[n_] := Module[{x, id = IntegerDigits[n, 2] // Reverse}, !IrreduciblePolynomialQ[id.x^Range[0, Length[id]-1], Modulus -> 2]];
Select[Range[2, 200], okQ] (* Jean-François Alcover, Jan 04 2022 *)

Edited by M. F. Hasler, Aug 18 2014

A234742 Product of the binary encodings of the irreducible factors (with multiplicity) of the polynomial over GF(2) whose encoding is n.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 7, 8, 21, 18, 11, 12, 13, 14, 27, 16, 81, 42, 19, 36, 49, 22, 39, 24, 25, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 75, 44, 189, 78, 47, 48, 77, 50, 243, 52, 57, 126, 55, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441

Offset: 0

Antti Karttunen, Jan 22 2014

"Product" refers to the ordinary multiplication of integers.
Differs from A235042 and A236837 for the first time at n=25, where a(n)=25, while A235042(25)=5 and A236837(25)=0. Thus A234741(A234742(n)) = n up to n=24.
a(n) >= n. [All terms of the table A061858 are nonnegative as the product of multiplying two numbers with carries is never less than when multiplying them without carries.]
Specifically, for all n, a(A091209(n)) > A091209(n).
a(A091209(n)) is always composite and, by the above inequality, larger than A091209(n), which implies that none of the terms of A091209 occur in this sequence. Cf. also A236844.
Starting with various terms (primes) in A235033 and iterating the map A234742, we get 5 -> 9 -> 21 -> 49 -> 77 -> 177 -> 333 = a(333).
Another example: 17 -> 81 -> 169 -> 309 -> 721 = a(721).
Does every chain of such iterations eventually reach a fixed point? (One of the terms of A235035.) Or do some of them manage to avoid such "traps" indefinitely? (Note how the terms of A235035 seem to get rarer, but only rather slowly.)
Starting from 23, we get the sequence: 23, 39, 99, 279, 775, 1271, 3003, 26411, 45059, ... which reaches its fixed point, 3643749709604450870616156947649219, after 55 iterations. - M. F. Hasler, Feb 18 2014. [This is now sequence A244323. See also A260729, A260735 and A260441.] - Antti Karttunen, Aug 05 2015
Note also that when coming backwards from some term of such a chain by iterating A234741, we may not necessarily end at the same term we started from.

			3 has binary representation '11', which encodes the polynomial X + 1, which is irreducible in GF(2)[X], so the result is just a(3)=3.
5 has binary representation '101' which encodes the polynomial X^2 + 1, which is reducible in the polynomial ring GF(2)[X], factoring as (X+1)(X+1), i.e., 5 = A048720(3,3), as 3 ('11' in binary) encodes the polynomial (X+1), irreducible in GF(2)[X]. 3*3 = 9, thus a(5)=9.
9 has binary representation '1001', which encodes the polynomial X^3 + 1, which factors (in GF(2)[X]!) as (X+1)(X^2+X+1), i.e., 9 = A048720(3,7) (7, '111' in binary, encodes the other factor polynomial X^2+X+1). 3*7 = 21, thus a(9)=21.
25 has binary representation '11001', which encodes the polynomial X^4 + X^3 + 1, which is irreducible in GF(2)[X], so the result is just a(25)=25.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

A235035 gives the k for which a(k)=k.
A236853(n) gives the number of times n occurs in this sequence.
A236842 gives the same sequence sorted and with duplicates removed, A236844 gives the numbers that do not occur here, A236845 gives numbers that occur more than once, A236846 the least inverse and A236847 the greatest inverse. A236850 gives such k that a(k) = A236837(k).
Cf. A014580, A048720, A234741, A236379, A236837, A236851 & A236852, A115857, A115872.
Cf. also A260712, A260713, A260716 and A244323, A260729, A260735, A260441 (iterations starting from various terms of A236844).

A234742(n)=factorback(subst(lift(factor(Mod(1,2)*Pol(binary(n)))),x,2)) \\ M. F. Hasler, Feb 18 2014, corrected Andrew Howroyd, Aug 01 2018

To compute a(n): factor the polynomial over GF(2) encoded by n, into its irreducible factors; in other words, find a unique multiset of terms i, j, ..., k (not necessarily distinct) from A014580 for which i x j x ... x k = n, where x stands for the carryless multiplication A048720. Then a(n) = i*j*...*k is the product of those terms with ordinary multiplication. Because of the effect of the carry-bits in the latter, the result is always greater than or equal to n, so we have a(n) >= n for all n.
a(2n) = 2*a(n).
a(A235035(n)) = A235035(n).
A236379(n) = a(n) - n.
For all n, a(n) >= A236837(n).

A091206 Primes whose binary representation encodes a polynomial irreducible over GF(2).

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 97, 103, 109, 131, 137, 157, 167, 191, 193, 211, 229, 239, 241, 283, 313, 379, 397, 419, 433, 463, 487, 499, 557, 563, 587, 601, 607, 613, 617, 631, 647, 661, 677, 701, 719, 757, 761, 769, 787, 827, 859

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).

Subsequence with Hamming weight nonprime starts 2, 1019, 1279, 1531, 1663, 1759, 1783, 1789, 2011, 2027, 2543, 2551, ... [Joerg Arndt, Nov 01 2013]. These are now given by A255569. - Antti Karttunen, May 14 2015

Antti Karttunen, Table of n, a(n) for n = 1..10226; all terms up to 1048357, binary length 20
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences related to binary encoded polynomials over GF(2)

Intersection of A014580 and A000040.
Apart from a(2) = 3 a subsequence of A027697. The numbers in A027697 but not here are listed in A238186.
Also subsequence of A235045 (its primes. Cf. also A235041-A235042).
Cf. A091209 (Primes whose binary expansion encodes a polynomial reducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)), A257688 (either 1, prime or irreducible over GF(2)).
Subsequence: A255569.

okQ[p_] := Module[{id, pol, x}, id = IntegerDigits[p, 2] // Reverse; pol = id.x^Range[0, Length[id] - 1]; IrreduciblePolynomialQ[pol, Modulus -> 2]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Feb 06 2023 *)

is(n)=polisirreducible( Mod(1,2) * Pol(digits(n,2)) );
forprime(n=2,10^3,if (is(n), print1(n,", ")));
\\ Joerg Arndt, Nov 01 2013

a(n) = A000040(A091207(n)) = A014580(A091208(n)).

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

Original entry on oeis.org

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

A235042 Factorization-preserving bijection from GF(2)[X]-polynomials to nonnegative integers, version which fixes the elements that are irreducible in both semirings.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 6, 7, 8, 21, 18, 11, 12, 13, 14, 27, 16, 81, 42, 19, 36, 49, 22, 39, 24, 5, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 15, 44, 189, 78, 47, 48, 77, 10, 243, 52, 57, 126, 17, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Like A091203 this is a factorization-preserving isomorphism from GF(2)[X]-polynomials to integers. The former are encoded in the binary representation of n like this: n=11, '1011' in binary, stands for polynomial x^3+x+1, n=25, '11001' in binary, stands for polynomial x^4+x^3+1. However, this version does not map the irreducible GF(2)[X] polynomials (A014580) straight to the primes (A000040), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A014580 \ A000040 (= A091214) in numerical order to the set-wise difference A000040 \ A014580 (= A091209).
The composite values are defined by the multiplicativity. E.g., we have a(A048724(n)) = 3*a(n) and a(A001317(n)) = A000244(n) = 3^n for all n.
This map satisfies many of the same identities as A091203, e.g., we have A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)) and A091247(n) = A066247(a(n)) for all n >= 1.

			Here (t X u) = A048720(t,u):
a(2)=2, a(3)=3 and a(7)=7, as 2, 3 and 7 are all in A091206.
a(4) = a(2 X 2) = a(2)*a(2) = 2*2 = 4.
a(5) = a(3 X 3) = a(3)*a(3) = 3*3 = 9.
a(9) = a(3 X 7) = a(3)*a(7) = 3*7 = 21.
a(10) = a(2 X 3 X 3) = a(2)*a(3)*a(3) = 2*3*3 = 18.
a(15) = a(3 X 3 X 3) = a(3)^3 = 3^3 = 27.
a(17) = a(3 X 3 X 3 X 3) = a(3)^4 = 3^4 = 81.
a(21) = a(7 X 7) = a(7)*a(7) = 7*7 = 49.
a(25) = 5, as 25 is the first term of A091214 and 5 is the first term of A091209.
a(50) = a(2 X 25) = a(2)*a(25) = 2*5 = 10.

Inverse: A235041. Fixed points: A235045.
Cf. A010051, A004247, A091225, A091209, A235044.
Similar cross-multiplicative permutations: A091203, A091205, A106443, A106445, A106447.

a(0)=0, a(1)=1, a(p) = p for those irreducible GF(2)[X]-polynomials whose binary encoding is a prime (i.e., p is in A091206), and for the rest of irreducible GF(2)[X]-polynomials (those which are encoded by a composite natural number, i.e., q is in A091214), a(q) = A091209(A235044(q)), and for reducible polynomials, a(i X j X k X ...) = a(i) * a(j) * a(k) * ..., where each i, j, k, ... is in A014580, X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).

A235041 Factorization-preserving bijection from nonnegative integers to GF(2)[X]-polynomials, version which fixes the elements that are irreducible in both semirings.

Original entry on oeis.org

0, 1, 2, 3, 4, 25, 6, 7, 8, 5, 50, 11, 12, 13, 14, 43, 16, 55, 10, 19, 100, 9, 22, 87, 24, 321, 26, 15, 28, 91, 86, 31, 32, 29, 110, 79, 20, 37, 38, 23, 200, 41, 18, 115, 44, 125, 174, 47, 48, 21, 642, 89, 52, 117, 30, 227, 56, 53, 182, 59, 172, 61, 62, 27, 64

Offset: 0

Antti Karttunen, Jan 02 2014

nonn

Like A091202 this is a factorization-preserving isomorphism from integers to GF(2)[X]-polynomials. The latter are encoded in the binary representation of n like this: n=11, '1011' in binary, stands for polynomial x^3+x+1, n=25, '11001' in binary, stands for polynomial x^4+x^3+1. However, this version does not map the primes (A000040) straight to the irreducible GF(2)[X] polynomials (A014580), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A000040 \ A014580 (= A091209) in numerical order to the set-wise difference A014580 \ A000040 (= A091214).
The composite values are defined by the multiplicativity. E.g., we have a(3n) = A048724(a(n)) and a(3^n) = A001317(n) for all n.
This map satisfies many of the same identities as A091202, e.g., we have A000005(n) = A091220(a(n)), A001221(n) = A091221(a(n)), A001222(n) = A091222(a(n)) and A008683(n) = A091219(a(n)) for all n >= 1.

			Here (t X u) = A048720(t,u):
a(2)=2, a(3)=3 and a(7)=7, as 2, 3 and 7 are all in A091206.
a(4) = a(2*2) = a(2) X a(2) = 2 X 2 = 4.
a(9) = a(3*3) = a(3) X a(3) = 3 X 3 = 5.
a(5) = 25, as 5 is the first term of A091209 and 25 is the first term of A091214.
a(10) = a(2*5) = a(2) X a(5) = 2 X 25 = 50.
Similarly, a(17) = 55, as 17 is the second term of A091209 and 55 is the second term of A091214.
a(21) = a(3*7) = a(3) X a(7) = 3 X 7 = 9.

Inverse: A235042. Fixed points: A235045.
Cf. A010051, A048720, A091225, A091214, A235043, A048724, A115857, A115872.
Similar cross-multiplicative permutations: A091202, A091204, A106442, A106444, A106446.

a(0)=0, a(1)=1, a(p) = p for those primes p whose binary representations encode also irreducible GF(2)[X]-polynomials (i.e., p is in A091206), and for the rest of the primes q (those whose binary representation encode composite GF(2)[X]-polynomials, i.e., q is in A091209), a(q) = A091214(A235043(q)), and for composite natural numbers, a(p * q * r * ...) = a(p) X a(q) X a(r) X ..., where p, q, r, ... are primes and X stands for the carryless multiplication (A048720) of GF(2)[X] polynomials encoded as explained in the Comments section.

A091212 Composite numbers whose binary representation encodes a polynomial reducible over GF(2).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99

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).
From Reinhard Zumkeller, Jul 05-12 2011, values for maximum n corrected by Antti Karttunen, May 18 2015: (Start)
a(n) = A192506(n) for n <= 36.
a(n) = A175526(n) for n <= 36.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..55429
A. Karttunen, Scheme-program for computing beginning of this sequence.
Index entries for sequences related to binary encoded polynomials over GF(2)

Intersection of A002808 and A091242.
Cf. A257688 (complement, either 1, irreducible in GF(2)[X] or prime), A091206 (prime and irreducible), A091209 (prime and reducible), A091214 (nonprime and irreducible).
Cf. A091213, A236861, A235036 (a subsequence, apart from 1).
Differs from both A175526 and A192506 for the first time at n=37, where a(37) = 56, while A175526(37) = A192506(37) = 55, a term missing from here (as 55 encodes a polynomial which is irreducible in GF(2)[X]).
Differs from its subsequence A205783(n+1) for the first time at n=47, where a(47) = 69, while 69 is missing from A205783.

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091212(n) = ((n > 1) && !isprime(n) && !isA014580(n));
n = 0; i = 0; while(n < 2^16, n++; if(isA091212(n), i++; write("b091212.txt", i, " ", n)));

a(n) = A091242(A091213(n)).

cp's OEIS Frontend

A235043 Partial sums of the characteristic function of A091209.

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A260716 a(n) = number of iterations of A234742 needed when starting from A091209(n) before a fixed point is reached.

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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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A091242 Reducible polynomials over GF(2), coded in binary.

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A234742 Product of the binary encodings of the irreducible factors (with multiplicity) of the polynomial over GF(2) whose encoding is n.

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A091206 Primes whose binary representation encodes a polynomial irreducible over GF(2).

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

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A235042 Factorization-preserving bijection from GF(2)[X]-polynomials to nonnegative integers, version which fixes the elements that are irreducible in both semirings.

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