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).

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

Author

Antti Karttunen, Aug 19 2014

Keywords

• nonn

Comments

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..195
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A246163.

Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.

Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

Formula

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].