A153151 Rotated binary decrementing: For n<2 a(n) = n, if n=2^k, a(n) = 2*n-1, otherwise a(n) = n-1.

0, 1, 3, 2, 7, 4, 5, 6, 15, 8, 9, 10, 11, 12, 13, 14, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 63, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 127, 64, 65, 66, 67, 68, 69

Offset: 0

Author

Antti Karttunen, Dec 20 2008

Keywords

• nonn
• base

Comments

Without the initial 0, a(n) is the lexicographically minimal sequence of distinct positive integers such that all values of a(n) mod n are distinct and nonnegative. - Ivan Neretin, Apr 27 2015

A002487(n)/A002487(n+1), n > 0, runs through all the reduced nonnegative rationals exactly once. A002487 is the Stern's sequence. Permutation from denominators (A002487(n+1))

1 2 1 3 2 3 1 4 3 5 2 5 3 4 1

where labels are

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

to numerators (A002487(n))

1 1 2 1 3 2 3 1 4 3 5 2 5 3 4

where changed labels are

1 3 2 7 4 5 6 15 8 9 10 11 12 13 14

Thus, b(n) = A002487(n+1), b(a(n)) = A002487(n), n>0. - Yosu Yurramendi, Jul 07 2016

Links

• A. Karttunen, Table of n, a(n) for n = 0..2047
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A153152.

Cf. A059893, A059894, A153141, A153142.

Programs

Maple

a := n -> if n < 2 then n elif convert(convert(n, base, 2), `+`) = 1 then 2*n-1 else n-1 fi: seq(a(n), n=0..70); # Peter Luschny, Jul 16 2016

Mathematica

Table[Which[n < 2, n, IntegerQ[Log[2, n]], 2 n - 1, True, n - 1], {n, 0, 70}] (* Michael De Vlieger, Apr 27 2015 *)

Python

def ok(n): return n&(n - 1)==0
def a(n): return n if n<2 else 2*n - 1 if ok(n) else n - 1 # Indranil Ghosh, Jun 09 2017

R

nmax <- 126 # by choice
a <- c(1,3,2)
for(n in 3:nmax) a[n+1] <- n
for(m in 0:floor(log2(nmax))) a[2^m] <- 2^(m+1) - 1
a <- c(0, a)
# Yosu Yurramendi, Sep 05 2020

Formula

a(n) = A059893(A153141(A059893(n))) = A059894(A153142(A059894(n))).