A014580 -id:A014580 - cp's OEIS frontend

A091225 Characteristic function of A014580: 1 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset: 0

Antti Karttunen, Jan 03 2004

nonn

Antti Karttunen, Table of n, a(n) for n = 0..100000
A. Karttunen, Scheme-program for computing this sequence
Index entries for characteristic functions
Index entries for sequences operating on GF(2)[X]-polynomials

a(n) = A010051(A091203(n)) = A010051(A091205(n)). Partial sums give A091226. Cf. A091227. Complementary to A091247.

a(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ Michel Marcus, Nov 11 2017

Data section extended up to a(120) by Antti Karttunen, Jan 01 2023

A245703 Permutation of natural numbers: a(1) = 1, a(p_n) = A014580(a(n)), a(c_n) = A091242(a(n)), where p_n = n-th prime, c_n = n-th composite number and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomials over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 11, 6, 8, 12, 25, 9, 13, 17, 10, 14, 47, 18, 19, 34, 15, 20, 31, 24, 16, 21, 62, 26, 55, 27, 137, 45, 22, 28, 42, 33, 37, 23, 29, 79, 59, 35, 87, 71, 36, 166, 41, 58, 30, 38, 54, 44, 61, 49, 32, 39, 99, 76, 319, 46, 91, 108, 89, 48, 200, 53, 97, 75, 40, 50, 203, 70, 67, 57, 78, 64, 43, 51

Offset: 1

Antti Karttunen, Aug 02 2014

All the permutations A091202, A091204, A106442, A106444, A106446, A235041 share the same property that primes (A000040) are mapped bijectively to the binary representations of irreducible GF(2) polynomials (A014580) but while they determine the mapping of composites (A002808) to the corresponding binary codes of reducible polynomials (A091242) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Inverse: A245704.
Cf. A000040, A002808, A000720, A007097, A010051, A014580, A065855, A091225, A091230, A091242.
Similar or related permutations: A091202, A091204, A106442, A106444, A106446, A235041, A135141, A245701, A245702, A245821, A245822, A244987, A245450.

allocatemem(123456789);
a014580 = vector(2^18);
a091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
A245703(n) = if(1==n, 1, if(isprime(n), a014580[A245703(primepi(n))], a091242[A245703(n-primepi(n)-1)]));
for(n=1, 10001, write("b245703.txt", n, " ", A245703(n)));

;; With memoization-macro definec.
(definec (A245703 n) (cond ((= 1 n) n) ((= 1 (A010051 n)) (A014580 (A245703 (A000720 n)))) (else (A091242 (A245703 (A065855 n))))))

a(1) = 1, a(p_n) = A014580(a(n)) and a(c_n) = A091242(a(n)), where p_n is the n-th prime, A000040(n) and c_n is the n-th composite, A002808(n).
a(1) = 1, after which, if A010051(n) is 1 [i.e. n is prime], then a(n) = A014580(a(A000720(n))), otherwise a(n) = A091242(a(A065855(n))).
As a composition of related permutations:
a(n) = A245702(A135141(n)).
a(n) = A091204(A245821(n)).
Other identities. For all n >= 1, the following holds:
a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A091204 has the same property]
A091225(a(n)) = A010051(n). [Maps primes to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242. The permutations A091202, A091204, A106442, A106444, A106446 and A235041 have the same property.]

A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 25, 14, 18, 19, 22, 26, 33, 38, 24, 11, 28, 30, 34, 39, 49, 23, 55, 36, 20, 42, 45, 37, 50, 56, 69, 47, 35, 77, 52, 32, 60, 17, 64, 54, 70, 78, 94, 66, 51, 29, 105, 74, 48, 41, 84, 53, 27, 88, 76, 95, 106, 73, 125, 91, 72, 44, 140, 97, 100, 68, 58, 115, 75, 40

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245703.
Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

allocatemem(123456789);
default(primelimit, 2^22)
A091226 = vector(2^22);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));

;; With memoization-macro definec.
(definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).
As a composition of related permutations:
a(n) = A227413(A245701(n)).
a(n) = A245822(A091205(n)).
Other identities. For all n >= 1, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

A091227 Inverse function of A014580: position in A014580 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 0, 0, 10, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 13, 0, 14, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 18, 0, 0, 0, 0, 0, 19, 0

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

Inverse of A014580. a(n) = A049084(A091203(n)).

a(n) = A091225(n) * A091226(n).

A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 8, 7, 11, 19, 6, 17, 10, 15, 23, 39, 13, 35, 18, 21, 31, 47, 79, 27, 16, 71, 37, 43, 63, 95, 14, 159, 55, 33, 143, 75, 22, 87, 127, 191, 38, 29, 319, 111, 67, 287, 12, 151, 45, 175, 255, 383, 77, 59, 34, 639, 223, 135, 20, 575, 30, 25, 303, 91, 351, 511, 46, 767, 155, 119, 69, 1279, 78, 447, 271, 41, 1151, 61, 51

Offset: 1

Antti Karttunen, Aug 02 2014

Antti Karttunen, Table of n, a(n) for n = 1..10001
Antti Karttunen, Entanglement Permutations, 2016-2017
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Inverse: A245702.
Cf. A000035, A014580, A091242, A091225, A091226, A091245.
Similar entanglement permutations: A135141, A193231, A237427, A243287, A245703, A245704.

allocatemem(123456789);
A091226 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245701(n) = if(1==n, 1, if(isA014580(n), 2*(A245701(A091226[n])), 1 + 2*(A245701(A091245(n)))));
for(n=1, 10001, write("b245701.txt", n, " ", A245701(n)));

;; With memoizing definec-macro.
(definec (A245701 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (* 2 (A245701 (A091226 n)))) (else (+ 1 (* 2 (A245701 (A091245 n)))))))

a(1) = 1, and for n > 1, if n is in A014580, a(n) = 2*a(A091226(n)), otherwise a(n) = 1 + 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A135141(A245704(n)).
Other identities:
For all n >= 1, 1 - A000035(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (= A014580) to even numbers and the corresponding representations of reducible polynomials to odd numbers].

A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 2, 4, 3, 5, 11, 8, 7, 6, 13, 9, 47, 17, 31, 14, 25, 12, 19, 10, 59, 20, 37, 15, 319, 62, 87, 24, 185, 42, 61, 21, 137, 34, 55, 18, 97, 27, 41, 16, 415, 76, 103, 28, 229, 49, 67, 22, 3053, 373, 433, 79, 647, 108, 131, 33, 1627, 222, 283, 54, 425, 78, 109, 29, 1123, 166, 203, 45, 379, 71, 91, 26, 731, 121, 145, 36, 253, 53, 73, 23

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Inverse: A245701.
Cf. A000035, A014580, A091242, A091225.
Similar entanglement permutations: A193231, A227413, A237126, A243288, A245703, A245704.

allocatemem(123456789);
a014580 = vector(2^18);
a091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
A245702(n) = if(1==n, 1, if(0==(n%2), a014580[A245702(n/2)], a091242[A245702((n-1)/2)]));
for(n=1, 383, write("b245702.txt", n, " ", A245702(n)));

;; With memoizing definec-macro.
(definec (A245702 n) (cond ((< n 2) n) ((even? n) (A014580 (A245702 (/ n 2)))) (else (A091242 (A245702 (/ (- n 1) 2))))))

a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)).
As a composition of related permutations:
a(n) = A245703(A227413(n)).
Other identities:
For all n >= 1, 1 - A091225(a(n)) = A000035(n). [Maps even numbers to binary representations of irreducible GF(2) polynomials (= A014580) and odd numbers to the corresponding representations of reducible polynomials].

A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2a(n))+1, a(A091242(n)) = 2a(n), where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

Original entry on oeis.org

1, 3, 7, 2, 6, 14, 15, 4, 12, 28, 5, 30, 13, 8, 24, 56, 10, 60, 29, 26, 16, 48, 112, 20, 31, 120, 58, 52, 32, 96, 9, 224, 40, 62, 240, 116, 25, 104, 64, 192, 57, 18, 448, 80, 124, 480, 11, 232, 50, 208, 128, 384, 114, 36, 61, 896, 160, 248, 27, 960, 17, 22, 464, 100, 416, 256, 49, 768, 228, 72, 122, 1792, 113, 320, 496, 54, 1920, 34, 44

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Because 2 is the only even term in A014580, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
Note that for any value k in A246156, "Odd reducible polynomials over GF(2)": 5, 9, 15, 17, 21, 23, ..., a(k) will be even, and apart from 2, all other even numbers are mapped to some even number, so all those terms reside in infinite cycles. Furthermore, apart from 5 and 15, all of them reside in separate cycles. The infinite cycle containing 5 and 15 goes as: ..., 47, 11, 5, 6, 14, 8, 4, 2, 3, 7, 15, 24, 20, 26, 120, 7680, ... and it is only because a(2) = 3, that it can temporarily switch back from even terms to odd terms, until after a(15) = 24 it is finally doomed to the eternal evenness.
(Compare also to the comments given at A246161).

Inverse: A246202.
Similar or related permutations: A245701, A246161, A006068, A054429, A193231, A246163, A246203, A237427.
Cf. A091225, A091226, A091245.

a(1) = 1, and for n > 1, if A091225(n) = 1 [i.e. when n is in A014580], a(n) = 1 + (2*a(A091226(n))), otherwise a(n) = 2*a(A091245(n)).
As a composition of related permutations:
a(n) = A054429(A245701(n)).
a(n) = A006068(A246161(n)).
a(n) = A193231(A246163(n)).
a(n) = A246203(A193231(n)).
Other identities:
For all n > 1, A000035(a(n)) = A091225(n). [After 1 maps binary representations of reducible GF(2) polynomials to even numbers and the corresponding representations of irreducible polynomials to odd numbers, in some order. A246203 has the same property].

A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

Original entry on oeis.org

1, 4, 2, 8, 11, 5, 3, 14, 31, 17, 47, 9, 13, 6, 7, 21, 61, 42, 185, 24, 87, 62, 319, 15, 37, 20, 59, 10, 19, 12, 25, 29, 109, 78, 425, 54, 283, 222, 1627, 33, 131, 108, 647, 79, 433, 373, 3053, 22, 67, 49, 229, 28, 103, 76, 415, 16, 41, 27, 97, 18, 55, 34, 137, 39, 167, 134, 859, 98, 563, 494, 4225, 70, 375, 331, 2705, 264, 2011, 1832, 19891, 44

Offset: 1

Antti Karttunen, Aug 19 2014

This sequence can be represented as a binary tree. Each left hand child is produced as A091242(n), and each right hand child as A014580(n), when the parent contains n:
                                     |
                  ...................1...................
                 4                                       2
       8......../ \.......11                   5......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  14       31         17       47          9       13          6       7
21  61   42  185    24  87   62  319     15 37   20  59      10 19   12 25
etc.
Because 2 is the only even term in A014580, it implies that, apart from a(3)=2, all other odd positions contain an odd number. There are also odd numbers in the even bisection, precisely all the terms of A246156 in some order, together with all even numbers larger than 2 that are also there. See also comments in A246201.

Inverse: A246201.
Similar or related permutations: A245702, A246162, A246164, A246204, A237126, A003188, A054429, A193231, A260422, A260426.
Cf. A014580, A091242, A091225, A246156.

allocatemem((2^31)+(2^30));
uplim = (2^25) + (2^24);
v014580 = vector(2^24);
v091242 = vector(uplim);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < uplim), if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++)
A246202(n) = if(1==n, 1, if(0==(n%2), v091242[A246202(n/2)], v014580[A246202((n-1)/2)]));
for(n=1, 638, write("b246202.txt", n, " ", A246202(n)));
\\ Works with PARI Version 2.7.4. - Antti Karttunen, Jul 25 2015
(Scheme, with memoization-macro definec)
(definec (A246202 n) (cond ((< n 2) n) ((odd? n) (A014580 (A246202 (/ (- n 1) 2)))) (else (A091242 (A246202 (/ n 2))))))

a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)).
As a composition of related permutations:
a(n) = A245702(A054429(n)).
a(n) = A246162(A003188(n)).
a(n) = A193231(A246204(n)).
a(n) = A246164(A193231(n)).
a(n) = A260426(A260422(n)).
Other identities:
For all n > 1, A091225(a(n)) = A000035(n). [After 1, maps even numbers to binary representations of reducible GF(2) polynomials and odd numbers to the corresponding representations of irreducible polynomials, in some order. A246204 has the same property].

A260426 a(1) = 1, a(A206074(n)) = A014580(a(n)), a(A205783(1+n)) = A091242(a(n)), where A014580 [respectively A091242] give binary codes for irreducible [resp. reducible] polynomials over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 11, 6, 8, 12, 25, 9, 13, 17, 10, 14, 47, 18, 19, 34, 15, 20, 31, 24, 55, 16, 21, 62, 137, 26, 37, 27, 45, 22, 28, 42, 59, 33, 71, 23, 87, 29, 41, 79, 166, 35, 61, 49, 36, 58, 30, 38, 319, 54, 91, 76, 44, 89, 97, 32, 203, 108, 39, 53, 99, 200, 67, 46, 103, 78, 185, 64, 131, 48, 75, 40, 379, 50, 73, 373, 109, 70, 433, 113, 95, 57, 1123, 111, 143, 121

Offset: 1

Antti Karttunen, Jul 26 2015

Each term of A260427 resides in a separate infinite cycle. This follows because any polynomial with (coefficients 0 or 1) that is irreducible over GF(2) is also irreducible over Q, in other words, A014580 is a subset of A206074. [See Thomas Ordowski's Feb 21 2014 comment in A014580] and thus any term of A091242 in A206074 is trapped into a trajectory containing only terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..7968
Index entries for sequences that are permutations of the natural numbers

Inverse: A260425.
Related permutations: A246202, A245703, A260421, A260424.
Cf. A014580, A091242, A205783, A206074, A257000, A255572, A255574, A260427.
Differs from A245703 for the first time at n=25, where a(25)=55, while A245703(25)=16.

allocatemem(234567890);
vecsize = (2^24)-4;
uplim = 2^25;
v014580 = vector(vecsize); A014580 = n -> v014580[n];
v091242 = vector(vecsize); A091242 = n -> v091242[n];
v255574 = vector(vecsize); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA206074(n) = polisirreducible(Pol(binary(n)));
A257000 = n -> isA206074(n);
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); v255574[n] = v255574[n-1]+A257000(n); if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++); print(n);
A260426(n) = if(1==n, 1, if(isA206074(n), A014580(A260426(A255574(n))), A091242(A260426(A255572(n)))));
for(n=1, 7968, write("b260426.txt", n, " ", A260426(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A014580(a(A255574(n))), otherwise [when n is in A205783], a(n) = A091242(a(A255572(n))).
As a composition of related permutations:
a(n) = A246202(A260421(n)).
a(n) = A245703(A260424(n)).

A246163 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A065621(1+a(n)), a(A091242(n)) = A048724(a(n)), where A065621(n) and A048724(n) give the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 3, 6, 9, 8, 5, 10, 27, 4, 24, 11, 15, 30, 45, 12, 40, 26, 29, 17, 34, 119, 20, 25, 120, 46, 39, 51, 102, 14, 153, 60, 43, 136, 114, 31, 105, 85, 170, 44, 18, 427, 68, 125, 408, 13, 150, 33, 187, 255, 510, 116, 54, 41, 765, 204, 135, 28, 680, 16, 23, 442, 99, 461, 257, 35, 514, 156, 90, 123, 1799, 118, 340, 393, 36

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)) and A065621/A048724, the latter which themselves are permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246161.
Because 3 is the only evil number in A014580, it implies that, apart from a(3)=7, odious numbers occur in odious positions only (along with many evil numbers that also occur in odious positions).
Furthermore, all terms of A246158 reside in infinite cycles, and apart from 4 and 8, all of them reside in separate cycles. The infinite cycle containing 4 and 8 goes as: ..., 2091, 97, 47, 13, 11, 4, 3, 7, 8, 5, 6, 9, 10, 27, 46, 408, 2535, ... and it is only because a(3) = 7, that it can temporarily switch back from evil terms to odious terms, until right after a(8) = 5 it is finally doomed to the eternal evilness.
Please see also the comments at A246201 and A246161.

Inverse: A246164.
Similar or related permutations: A246205, A193231, A246201, A234026, A245701, A234612, A246161, A246203.
Cf. A010060, A014580, A048724, A065621, A091225, A091226, A091245.

a(1) = 1, and for n > 1, if n is in A014580, a(n) = A065621(1+a(A091226(n))), otherwise a(n) = A048724(a(A091245(n))).
As a composition of related permutations:
a(n) = A193231(A246201(n)).
a(n) = A234026(A245701(n)).
a(n) = A234612(A246161(n)).
a(n) = A193231(A246203(A193231(n))).
Other identities:
For all n > 1, A010060(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (A014580) to odious numbers and the corresponding representations of reducible polynomials (A091242) to evil numbers, in some order].

cp's OEIS Frontend

A091225 Characteristic function of A014580: 1 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

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A245703 Permutation of natural numbers: a(1) = 1, a(p_n) = A014580(a(n)), a(c_n) = A091242(a(n)), where p_n = n-th prime, c_n = n-th composite number and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomials over GF(2), respectively.

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A091227 Inverse function of A014580: position in A014580 if the n-th GF(2)[X] polynomial is irreducible, 0 otherwise.

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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2*a(n), a(A091242(n)) = 2*a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

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A245702 Permutation of natural numbers: a(1) = 1, a(2n) = A014580(a(n)), a(2n+1) = A091242(a(n)), where A014580(n) = binary code for n-th irreducible polynomial over GF(2) and A091242(n) = binary code for n-th reducible polynomial over GF(2).

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A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2*a(n))+1, a(A091242(n)) = 2*a(n), where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

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A246202 Permutation of natural numbers: a(1) = 1, a(2n) = A091242(a(n)), a(2n+1) = A014580(a(n)), where A091242(n) = binary code for n-th reducible polynomial over GF(2) and A014580(n) = binary code for n-th irreducible polynomial over GF(2).

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A245701 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2a(n), a(A091242(n)) = 2a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2a(n))+1, a(A091242(n)) = 2a(n), where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).