

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).


7



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
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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 memoizationmacro 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 A011304 A196392
Adjacent sequences: A246160 A246161 A246162 * A246164 A246165 A246166


KEYWORD

nonn


AUTHOR

Antti Karttunen, Aug 19 2014


STATUS

approved



