A258996 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A162911/A162912 (Drib), and vice versa.

1, 2, 3, 6, 7, 4, 5, 10, 11, 8, 9, 14, 15, 12, 13, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 42, 43, 40, 41, 46, 47, 44, 45, 34, 35, 32, 33, 38, 39, 36, 37, 58, 59, 56, 57, 62, 63, 60, 61, 50, 51, 48, 49, 54, 55, 52, 53

Offset

1,2

Comments

As A258746 the permutation is self-inverse. Except for fixed points 1, 2, 3 it consists completely of 2-cycles: (4,6), (5,7), (8,10), (9,11), (12,14), (13,15), (16,26), (17,27), ..., (21,31), ..., (32,42), ... . - Yosu Yurramendi, Mar 31 2016

When terms of sequence |n - a(n)|/2 (n > 3) are considered only once, and they are sorted in increasing order, A147992 is obtained. - Yosu Yurramendi, Apr 05 2016

Links

Yosu Yurramendi, Table of n, a(n) for n = 1..16383

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1, a(2) = 2, a(3) = 3. For n = 2^m + k, m > 1, 0 <= k < 2^m. If m is even, then a(2^(m+1)+k) = a(2^m + k) + 2^m and a(2^(m+1) + 2^m+k) = a(2^m+k) + 2^(m+1). If m is odd, then a(2^(m+1) + k) = a(2^m+k) + 2^(m+1) and a(2^(m+1) + 2^m+k) = a(2^m+k) + 2^m.

From Yosu Yurramendi, Mar 23 2017: (Start)

A258746(a(n)) = a(A258746(n)), n > 0.

A092569(a(n)) = a(A092569(n)), n > 0.

A117120(a(n)) = a(A117120(n)), n > 0;

A065190(a(n)) = a(A065190(n)), n > 0;

A054429(a(n)) = a(A054429(n)), n > 0;

A063946(a(n)) = a(A063946(n)), n > 0. (End)

a(1) = 1, for m >= 0 and 0 <= k < 2^m, a(2^(m+1) + 2*k) = 2*a(2^(m+1) - 1 - k), a(2^(m+1) + 2*k + 1) = 2*a(2^(m+1) - 1 - k) + 1. - Yosu Yurramendi, May 23 2020

Prog

(R)
maxlevel <- 5 # by choice
a <- 1
for(m in 0:maxlevel) for(k in 0:(2^m-1)){
  a[2^(m+1) + 2*k    ] = 2*a[2^(m+1) - 1 - k]
  a[2^(m+1) + 2*k + 1] = 2*a[2^(m+1) - 1 - k] + 1}
a
(R) # Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # by choice
a <- 1:3
for(n in 4:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  anbit[seq(2, length(anbit) - 1, 2)] <- 1 - anbit[seq(2, length(anbit) - 1, 2)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Mar 30 2021

Crossrefs

Cf. A092569, A117120, A258746. Similar R-programs: A332769, A284447.

Sequence in context: A169746 A368160 A234613 * A092569 A361996 A191726

Adjacent sequences: A258993 A258994 A258995 * A258997 A258998 A258999

Keyword

nonn,look

Author

Yosu Yurramendi, Jun 16 2015

Status

approved