A246204 -id:A246204 - cp's OEIS frontend

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).

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].

A246164 Permutation of natural numbers: a(1) = 1, a(A065621(n)) = A014580(a(n-1)), a(A048724(n)) = A091242(a(n)), where A065621(n) and A048724(n) are 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, 4, 11, 8, 5, 3, 7, 6, 9, 13, 17, 47, 31, 14, 61, 21, 42, 185, 24, 87, 319, 62, 12, 25, 19, 10, 59, 20, 15, 37, 229, 49, 22, 67, 76, 415, 103, 28, 18, 55, 137, 34, 41, 16, 27, 97, 78, 425, 109, 29, 1627, 222, 54, 283, 433, 79, 373, 3053, 33, 131, 647, 108, 847, 133, 745, 6943, 44, 193, 1053, 160, 504, 4333, 587, 99

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 A065621/A048724 and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).
The former are themselves permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246162.
For the comments about the cycle structure, please see A246163.

Inverse: A246163.
Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.
Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. when n is an odious number], a(n) = A014580(a(A065620(n)-1)), otherwise a(n) = A091242(a(- (A065620(n)))). [A065620 Converts sum of powers of 2 in binary representation of n to an alternating sum].
As a composition of related permutations:
a(n) = A246202(A193231(n)).
a(n) = A245702(A234025(n)).
a(n) = A246162(A234612(n)).
a(n) = A193231(A246204(A193231(n))).
For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order. A246162 has the same property].

A246203 Permutation of natural numbers: a(n) = A246201(A193231(n)).

Original entry on oeis.org

1, 7, 3, 6, 2, 14, 15, 24, 8, 30, 13, 28, 5, 12, 4, 10, 56, 60, 29, 26, 16, 112, 48, 96, 9, 32, 52, 58, 120, 20, 31, 128, 208, 232, 50, 36, 61, 114, 384, 960, 17, 464, 22, 160, 896, 248, 27, 62, 240, 40, 224, 64, 104, 116, 25, 124, 80, 480, 11, 192, 57, 448, 18, 1536, 98, 456, 21, 928, 200, 512, 832, 3584, 121, 244, 144

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

This permutation has the same cycle structure as A246163 has because this is its A193231-conjugate.
On the other hand, it shares with A246201 the following property:
Because 2 is the only even term in A014580, it implies that, apart from a(2)=7, 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, and apart from 5 and 15, all of them reside in separate cycles. The infinite cycle containing 5 and 15 goes as: ..., 14523, 3889, 103, 59, 11, 13, 5, 2, 7, 15, 4, 6, 14, 12, 28, 58, 480, 3728, 3932416, ... and it is only because a(2) = 7, that it can temporarily switch back from even terms to odd terms, until right after a(15) = 4 it is finally doomed to the eternal evenness.
See also comments at A246161 and A246163.

Inverse: A246204.
Related permutations: A193231, A246201, A246161, A246163.
Cf. also A000035, A091225, A246156.

(define (A246203 n) (A246201 (A193231 n)))

a(n) = A246201(A193231(n)).
a(n) = A193231(A246163(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].

cp's OEIS Frontend

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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A246203 Permutation of natural numbers: a(n) = A246201(A193231(n)).

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