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).

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

1,2

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10001

Index entries for sequences operating on GF(2)[X]-polynomials

Index entries for sequences that are permutations of the natural numbers

Formula

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].

Prog

(Scheme, with memoization-macro definec)
(definec (A246163 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A065621 (+ 1 (A246163 (A091226 n))))) (else (A048724 (A246163 (A091245 n))))))

Crossrefs

Inverse: A246164.

Similar or related permutations: A246205, A193231, A246201, A234026, A245701, A234612, A246161, A246203.

Cf. A010060, A014580, A048724, A065621, A091225, A091226, A091245.

Sequence in context: A104957 A329333 A083119 * A198388 A334375 A011304

Adjacent sequences: A246160 A246161 A246162 * A246164 A246165 A246166

Keyword

nonn

Author

Antti Karttunen, Aug 19 2014

Status

approved