A332769 -id:A332769 - cp's OEIS frontend

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

Yosu Yurramendi, Jun 16 2015

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

Yosu Yurramendi, Table of n, a(n) for n = 1..16383
Index entries for sequences that are permutations of the natural numbers

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

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

# 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

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
a(n) = A020988(A102572(n)) XOR n. - Alan Michael Gómez Calderón, Mar 11 2025

A153154 Permutation of natural numbers: A059893-conjugate of A006068.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 5, 6, 15, 8, 9, 14, 11, 12, 13, 10, 31, 16, 17, 30, 19, 28, 29, 18, 23, 24, 25, 22, 27, 20, 21, 26, 63, 32, 33, 62, 35, 60, 61, 34, 39, 56, 57, 38, 59, 36, 37, 58, 47, 48, 49, 46, 51, 44, 45, 50, 55, 40, 41, 54, 43, 52, 53, 42, 127, 64, 65, 126, 67, 124

Offset: 0

Antti Karttunen, Dec 20 2008

A002487(1+a(n)) = A020651(n) and A002487(a(n)) = A020650(n). So, it generates the enumeration system of positive rationals based on Stern's sequence A002487. - Yosu Yurramendi, Feb 26 2020

Inverse: A153153. a(n) = A059893(A006068(A059893(n))).

maxn <- 63 # by choice
a <- c(1,3,2)
#
for(n in 2:maxn){
  a[2*n] <- 2*a[n] + 1
  if(n%%2==0) a[2*n+1] <- 2*a[n+1]
  else        a[2*n+1] <- 2*a[n-1]
}
(a <- c(0,a))
# Yosu Yurramendi, Feb 26 2020

# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 8 # by choice
a <- c(1, 3, 2)
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
  for(i in 2:(length(anbit) - 1))
    anbit[i] <- bitwXor(anbit[i], anbit[i - 1])  # ?bitwXor
  anbit[0:(length(anbit) - 1)] <- 1 - anbit[0:(length(anbit) - 1)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
(a <- c(0, a))
# Yosu Yurramendi, Oct 04 2021

From Yosu Yurramendi, Feb 26 2020: (Start)
a(1) = 1, for all n > 0 a(2*n) = 2*a(n) + 1, a(2*n+1) = 2*a(A065190(n)).
a(1) = 1, a(2) = 3, a(3) = 2, for all n > 1 a(2*n) = 2*a(n) + 1, and if n even a(2*n+1) = 2*a(n+1), else a(2*n+1) = 2*a(n-1).
a(n) = A054429(A231551(n)) = A231551(A065190(n)) = A284459(A054429(n)) =
       A332769(A284459(n)) = A258996(A154437(n)). (End)

A284447 Permutation of the positive integers: a(n) = A258996(A092569(n)) = A092569(A258996(n)).

Original entry on oeis.org

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

Offset: 1

Yosu Yurramendi, Apr 06 2017

nonn

The permutation is self-inverse. Except for fixed points 1, 2, 3, 4, 5, 6, 7 it consists completely of 2-cycles: (8,12), (9,13), (10,14), (11,15), (16,20), (17,21), (18,22), (19,23), (24,28), (25,29), (26,30), (27,31), (32,52), (33,53), (34,54), (35,55), (36,48), (37,49), (38,50), (39,51), (40,60), ...

{A000027, A258996, A092569, a = A258996(A092569)} form a Klein 4-group.

Yosu Yurramendi, Table of n, a(n) for n = 1..32767

Analogous to A284120. Similar R-programs: A258996, A332769.

maxrow <- 8 # by choice
a <- 1:3
for(m in 1:maxrow) for(k in 0:(2^m-1)){
if(m%%2 == 1){a[2^(m+1)+    k] <- a[2^m+k] + 2^m
              a[2^(m+1)+2^m+k] <- a[2^m+k] + 2^(m+1)}
else         {a[2^(m+1)+    k] <- a[2^m+k] + 2^(m+1)
              a[2^(m+1)+2^m+k] <- a[2^m+k] + 2^m}
}
a
# Yosu Yurramendi, Apr 06 2017

# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # by choice
a <- 1:7
for(n in 8:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  anbit[seq(3, length(anbit) - 1, 2)] <- 1 - anbit[seq(3, length(anbit) - 1, 2)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Mar 30 2021

cp's OEIS Frontend

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.

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A153154 Permutation of natural numbers: A059893-conjugate of A006068.

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A284447 Permutation of the positive integers: a(n) = A258996(A092569(n)) = A092569(A258996(n)).

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