A080542 In binary representation: keep the first digit and rotate right the others.

1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 13, 10, 14, 11, 15, 16, 24, 17, 25, 18, 26, 19, 27, 20, 28, 21, 29, 22, 30, 23, 31, 32, 48, 33, 49, 34, 50, 35, 51, 36, 52, 37, 53, 38, 54, 39, 55, 40, 56, 41, 57, 42, 58, 43, 59, 44, 60, 45, 61, 46, 62, 47, 63, 64, 96, 65, 97, 66, 98, 67, 99, 68

Offset

1,2

Comments

Permutation of natural numbers with inverse = A080541: A080541(a(n)) = a(A080541(n)) = n;

let r(n,0)=n, r(n,k)=a(r(n,k-1)) for k>0, then r(n,floor(log_2(n))) = n and for n>1: r(n,floor(log_2(n))-1) = A080541(n).

Discarding their most significant bit, binary representations of numbers present in each cycle of this permutation form a distinct equivalence class of binary necklaces, thus there are A000031(n) separate cycles in each range [2^n .. (2^(n+1))-1] (for n >= 0) of this permutation. A256999 gives the largest number present in n's cycle. - Antti Karttunen, May 16 2015

Links

Antti Karttunen, Table of n, a(n) for n = 1..8192

Index entries for sequences related to necklaces

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = 2^log2(n) + floor((n-2^log2(n))/2) + (n mod 2)*2^(log2(n)-1), where log2(n) is the integer part of base-2 logarithm.

From Antti Karttunen, May 16 2015: (Start)

a(1) = 1; for n > 1, a(n) = A053644(n) + (A000035(n)*A072376(n)) + A004526(A053645(n)). [Essentially the same formula but represented with A-numbers.]

Other identities. For all n >= 1:

a(n) = A059893(A080541(A059893(n))).

a(n) = A054429(a(A054429(n))).

(End)

Example

a(20) = a('10100') = '10010' = 18.
a(25) = a('11001') = '11100' = 28.

Mathematica

kfd[n_]:=Module[{a, b}, {a, b}=TakeDrop[IntegerDigits[n, 2], 1]; FromDigits[ Join[a, RotateRight[b]], 2]]; Array[kfd, 80] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Feb 12 2016 *)

Prog

(Scheme) (define (A080542 n) (if (< n 2) n (+ (A053644 n) (+ (* (A000035 n) (A072376 n)) (A004526 (A053645 n))))))  ;; Antti Karttunen, May 16 2015
(R)
nmax <- 31 # by choice
a <- 1:3
for(n in 1:nmax) for(k in 0:3)
a[4*n + k] = 2*a[2*n + (k == 1 | k == 3)] + (k == 2 | k == 3)
a
# Yosu Yurramendi, Sep 05 2020
(Python)
def A080542(n): return (1+(n&1))*(1<<n.bit_length()-2)+(n>>1) if n > 1 else n # Chai Wah Wu, Jan 22 2023

Crossrefs

Inverse: A080541.

Cf. A000031, A000035, A000523, A004526, A007088, A053644, A053645, A054429, A072376, A080414, A080544, A256999, A257250.

The set of permutations {A059893, A080541, A080542} generates an infinite dihedral group.

Sequence in context: A257683 A343152 A267105 * A330090 A072758 A104464

Adjacent sequences: A080539 A080540 A080541 * A080543 A080544 A080545

Keyword

nonn,base

Author

Reinhard Zumkeller, Feb 20 2003

Status

approved