A092569 -id:A092569 - cp's OEIS frontend

A092570 Primes p which become a prime q under transformation of inner bits of binary representation in A092569. In binary representation of p, transformation of inner bits, 1 <-> 0, gives binary representation of q.

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 43, 53, 59, 79, 83, 89, 103, 109, 113, 151, 157, 173, 191, 193, 211, 227, 233, 269, 277, 281, 307, 311, 337, 347, 349, 359, 367, 379, 389, 401, 409, 419, 421, 431, 457, 461, 487, 491, 499, 523, 569, 599, 607, 617, 653, 659

Offset: 1

Zak Seidov, Feb 28 2004

			307 and 461 are terms because 307_10 = 100110011_2, transformation of inner bits gives 100110011_2 -> 111001101_2 = 461_10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A092569.

ptibQ[n_]:=Module[{id=IntegerDigits[n,2],f,l,r},f=id[[1]];l=id[[-1]];r=Most[Rest[id]];PrimeQ[FromDigits[Join[{f},r/.{1->0,0->1},{l}],2]]]; Select[Prime[Range[200]],ptibQ] (* Harvey P. Dale, Jul 16 2025 *)

T(p)={pow2=2;v=binary(p);L=#v-1;forstep(k=L,2,-1,if(v[k],p-=pow2,p+=pow2);pow2*=2);return(p)};
forprime(p=2,659,if(isprime(T(p)),print1(p,", ")))
\\ Washington Bomfim, Jan 18 2011

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

A054429 Simple self-inverse permutation of natural numbers: List each block of 2^n numbers (from 2^n to 2^(n+1) - 1) in reverse order.

Original entry on oeis.org

1, 3, 2, 7, 6, 5, 4, 15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 127, 126, 125, 124, 123, 122, 121

Offset: 1

Antti Karttunen

a(n) gives the position of the inverse of the n-th term in the full Stern-Brocot tree: A007305(a(n)+2) = A047679(n) and A047679(a(n)) = A007305(n+2). - Reinhard Zumkeller, Dec 22 2008
From Gary W. Adamson, Jun 21 2012: (Start)
The mapping and conversion rules are as follows:
By rows, we have ...
   1;
   3,  2;
   7,  6,  5,  4;
  15, 14, 13, 12, 11, 10, 9, 8;
  ... onto which we are to map one-half of the Stern-Brocot infinite Farey Tree:
  1/2
  1/3, 2/3
  1/4, 2/5, 3/5, 3/4
  1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5
  ...
The conversion rules are: Convert the decimal to binary, adding a duplicate of the rightmost binary term to its right. For example, 10 = 1010, which becomes 10100. Then, from the left, record the number of runs = [1,1,1,2], the continued fraction representation of 5/8. Check: 10 decimal corresponds to 5/8 as shown in the overlaid mapping. Take decimal 9 = 1001 which becomes 10011, with a continued fraction representation of [1,2,2] = 5/7. Check: 9 decimal corresponds to 5/7 in the Farey Tree map. (End)
From Indranil Ghosh, Jan 19 2017: (Start)
a(n) is the value generated when n is converted into its Elias gamma code, the 1's and 0's are interchanged and the resultant is converted back to its decimal value for all values of n > 1. For n = 1, A054429(n) = 1 but after converting 1 to Elias gamma code, interchanging the 1's and 0's and converting it back to decimal, the result produced is 0.
For example, let n = 10. The Elias gamma code for 10 is '1110010'. After interchanging the 1's and 0's it becomes "0001101" and 1101_2 = 13_10. So a(10) = 13. (End)
From Yosu Yurramendi, Mar 09 2017 (similar to Zumkeller's comment): (Start)
A002487(a(n)) = A002487(n+1), A002487(a(n)+1) = A002487(n), n > 0.
A162909(a(n)) = A162910(n), A162910(a(n)) = A162909(n), n > 0.
A162911(a(n)) = A162912(n), A162912(a(n)) = A162911(n), n > 0.
A071766(a(n)) = A245326(n), A245326(a(n)) = A071766(n), n > 0.
A229742(a(n)) = A245325(n), A245325(a(n)) = A229742(n), n > 0.
A020651(a(n)) = A245327(n), A245327(a(n)) = A020651(n), n > 0.
A020650(a(n)) = A245328(n), A245328(a(n)) = A020650(n), n > 0. (End)
From Yosu Yurramendi, Mar 29 2017: (Start)
A063946(a(n)) = a(A063946(n)) = A117120(n), n > 0.
A065190(a(n)) = a(A065190(n)) = A092569(n), n > 0.
A258746(a(n)) = a(A258746(n)) = A165199(n), n > 0.
A258996(a(n)) = a(A258996(n)), n > 0.
A117120(a(n)) = a(A117120(n)), n > 0.
A092569(a(n)) = a(A092569(n)), n > 0. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

See also A054424, A054430.
{A000027, A054429, A059893, A059894} form a 4-group.
Cf. A053644, A115303, A115304, A115305, A115306, A115307, A115308, A115309, A106649.
This is Guy Steele's sequence GS(6, 5) (see A135416).

a054429 n = a054429_list !! (n-1)
a054429_list = f [1..] where
   f xs@(x:_) = reverse us ++ f vs where (us, vs) = splitAt x xs
-- Reinhard Zumkeller, Jun 01 2015, Feb 21 2014

A054429 := n -> 3*2^ilog2(n) - n - 1:
seq(A054429(n), n = 1..70); # [Updated by Peter Luschny, Apr 24 2024]

Flatten[Table[Range[2^(n+1)-1,2^n,-1],{n,0,6}]] (* Harvey P. Dale, Dec 17 2013 *)

A054429(n)= 3<<#binary(n\2)-n-1 \\ M. F. Hasler, Aug 18 2014

from itertools import count, islice
def A054429_gen(): # generator of terms
    return (m for n in count(0) for m in range((1<A054429_list = list(islice(A054429_gen(),30)) # Chai Wah Wu, Jul 27 2023

maxblock <- 10 # by choice
a <- NULL
for(m in 0:maxblock) a <- c(a, rev(2^m:(2^(m+1)-1)))
a
# Yosu Yurramendi, Mar 10 2017

a(n) = ReflectBinTreePermutation(n).
a(n) = if n=1 then 1 else 2*a(floor(n/2)) + 1 - n mod 2. - Reinhard Zumkeller, Feb 18 2003
G.f.: 1/(1-x) * ((x-2x^2)/(1-x) + Sum_{k>=0} 3*2^k*x^2^k). - Ralf Stephan, Sep 15 2003
A000120(a(n)) = A000120(A059894(n)) = A023416(n) + 1. - Ralf Stephan, Oct 05 2003
A115310(n, 1) = a(n). - Reinhard Zumkeller, Jan 20 2006
a(1) = 1, 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, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Apr 06 2017
a(n) = A117120(A063946(n)) = A063946(A117120(n)) = A092569(A065190(n)) = A065190(A092569(n)), n > 0. - Yosu Yurramendi, Apr 10 2017
a(n) = 3*A053644(n) - n - 1. - Alan Michael Gómez Calderón, Feb 28 2025

A065190 Self-inverse permutation of the positive integers: 1 is fixed, followed by an infinite number of adjacent transpositions (n n+1).

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69, 68, 71, 70, 73

Offset: 1

Antti Karttunen, Oct 19 2001

Also, a lexicographically minimal sequence of distinct positive integers such that a(n) is coprime to n. - Ivan Neretin, Apr 18 2015
The larger term of the pair (a(n), a(n+1)) is always odd. Had we started the sequence with a(1) = 0, it would be the lexicographically first sequence with this property if always extented with the smallest integer not yet present. - Eric Angelini, Feb 17 2017
From Yosu Yurramendi, Mar 21 2017: (Start)
This sequence is self-inverse. Except for the fixed point 1, it consists completely of 2-cycles: (2n, 2n+1), n > 0.
A020651(a(n)) = A020650(n), A020650(a(n)) = A020651(n), n > 0.
A245327(a(n)) = A245328(n), A245328(a(n)) = A245327(n), n > 0.
A063946(a(n)) = a(A063946(n)), n > 0.
A054429(a(n)) = a(A054429(n)) = A092569(n), n > 0.
A258996(a(n)) = a(A258996(n)), n > 0.
A258746(a(n)) = a(A258746(n)), n > 0. (End)
From Enrique Navarrete, Nov 13 2017: (Start)
With a(0)=0, and the rest of the sequence appended, a(n) is the smallest positive number not yet in the sequence such that the arithmetic mean of the first n+1 terms a(0), a(1), ..., a(n) is not an integer; i.e., the sequence is 0, 1, 3, 2, 5, 4, 7, 6, 9, 8, ...
Example: for n=5, (0 + 1 + 3 + 2 + 5)/5 is not an integer.
Fixed points are odd numbers >= 3 and also a(n) = n-2 for even n >= 4. (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000
F. M. Dekking, Permutations of N generated by left-right filling algorithms, arXiv:2001.08915 [math.CO], 2020.
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004442, A065190 o A014681 = A065168, A014681 o A065190 = A065164.

[1] cat [n+(-1)^n: n in [2..80]]; // Vincenzo Librandi, Apr 18 2015

[seq(f(j),j=1..120)]; f := (n) -> `if`((n < 2), n,n+((-1)^n));

f[n_] := Rest@ Flatten@ Transpose[{Range[1, n + 1, 2], {1}~Join~Range[2, n, 2]}]; f@ 72 (* Michael De Vlieger, Apr 18 2015 *)
Rest@ CoefficientList[Series[x (x^3 - 2 x^2 + 2 x + 1)/((x - 1)^2*(x + 1)), {x, 0, 72}], x] (* Michael De Vlieger, Feb 17 2017 *)
Join[{1},LinearRecurrence[{1,1,-1},{3,2,5},80]] (* Harvey P. Dale, Feb 24 2021 *)

{ for (n=1, 1000, if (n>1, a=n + (-1)^n, a=1); write("b065190.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 13 2009

x='x+O('x^100); Vec(x*(x^3-2*x^2+2*x+1)/((x-1)^2*(x+1))) \\ Altug Alkan, Feb 04 2016

def a(n): return 1 if n<2 else n + (-1)**n # Indranil Ghosh, Mar 22 2017

maxrow <- 8 # by choice
a <- c(1,3,2) # If it were c(1,2,3), it would be A000027
  for(m in 1:maxrow) for(k in 0:(2^m-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)
}
a
# Yosu Yurramendi, Apr 10 2017

a(1) = 1, a(n) = n+(-1)^n.
From Colin Barker, Feb 18 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
G.f.: x*(x^3 - 2*x^2 + 2*x + 1) / ((x-1)^2*(x+1)). (End)
a(n)^a(n) == 1 (mod n). - Thomas Ordowski, Jan 04 2016
E.g.f.: x*(1+exp(x)) - 1 + exp(-x). - Robert Israel, Feb 04 2016
a(n) = A014681(n-1) + 1. - Michel Marcus, Dec 10 2016
a(1) = 1, for n > 0 a(2*n) = 2*a(a(n)) + 1, a(2*n + 1) = 2*a(a(n)). - Yosu Yurramendi, Dec 12 2020

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.

Original entry on oeis.org

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

A258746 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A007305/A047679 (Stern-Brocot) into the enumeration system A162909/A162910 (Bird), and vice versa.

Original entry on oeis.org

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

Offset: 1

Yosu Yurramendi, Jun 09 2015

nonn

As A117120 the permutation is self-inverse. Except for fixed points 1, 2, 3 it consists completely of 2-cycles: (4,5), (6,7), (8,10), (9,11), (12,14), (13,15), (16,21), (17,20), ..., (24,29), ..., (32,42), ... .

Yosu Yurramendi, Table of n, a(n) for n = 1..20001
Index entries for sequences that are permutations of the natural numbers

Cf. A117120.

a <- 1:3
maxn <- 50 # by choice
#
for(n in 2:maxn){
  m <- floor(log2(n))
  if(m%%2 == 0) {
    a[2*n  ] <- 2*a[n]
    a[2*n+1] <- 2*a[n]+1 }
  else {
    a[2*n  ] <- 2*a[n]+1
    a[2*n+1] <- 2*a[n]   }
}
#
a
# Yosu Yurramendi, Jun 09 2015

# 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
ifelse(floor(log2(n)) %% 2 == 0,
   anbit[seq(1, length(anbit)-1, 2)] <- 1 - anbit[seq(1, length(anbit)-1, 2)],
   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, May 29 2021

a(1) = 1, a(2) = 2, a(3) = 3. For n >= 2, m = floor(log_2(n)). If m even, then a(2*n) = 2*a(n) and a(2*n+1) = 2*a(n)+1. If m odd, then a(2*n) = 2*a(n)+1 and a(2*n+1) = 2*a(n).
From Yosu Yurramendi, Mar 23 2017: (Start)
A258996(a(n)) = a(A258996(n)) for n > 0;
A117120(a(n)) = a(A117120(n)) for n > 0;
A092569(a(n)) = a(A092569(n)) for n > 0;
A063946(a(n)) = a(A063946(n)) for n > 0;
A054429(a(n)) = a(A054429(n)) = A165199(n) for n > 0;
A065190(a(n)) = a(A065190(n)) for n > 0. (End)
a(n) = A054429(A165199(n)). - Alan Michael Gómez Calderón, Mar 08 2025

A117120 a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 10, 9, 8, 15, 14, 13, 12, 23, 22, 21, 20, 19, 18, 17, 16, 31, 30, 29, 28, 27, 26, 25, 24, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 95, 94, 93, 92, 91, 90, 89, 88, 87

Offset: 1

Leroy Quet, Apr 19 2006

Sequence is a permutation of the positive integers.
The permutation is self-inverse. Except for fixed points 1, 2, 3 it consists completely of 2-cycles: (4,5), (6,7), (8,11), (9,10), (12,15), (13,14), (16,23), (17,22), ..., (24,31), ..., (32,47), ... . - Klaus Brockhaus
The permutation transforms enumeration system of positive irreducible fractions A071766/A229742 (HCS) into enumeration system A245325/A245326, and vice versa. - Yosu Yurramendi, Jun 09 2015
A092569(a(n)) = a(A092569(n)), n > 0.
A258746(a(n)) = a(A258746(n)), n > 0.
A258996(a(n)) = a(A258996(n)), n > 0.
A054429(a(n)) = a(A054429(n)), n > 0.
a(n) = A054429(A063946(n)) = A063946(A054429(n)), n > 0. - Yosu Yurramendi, Mar 23 2017

Cf. A071766, A229742, A245325, A245326.

A[1]:= 1: A[2]:= 2: B[1]:= 0: B[2]:= 0:
for n from 3 to 100 do
  for m from A[n-1]-1 by A[n-1] while assigned(B[m]) do od:
  A[n]:= m;
  B[m]:= 0;
od:
seq(A[n],n=1..100); # Robert Israel, Jun 09 2015

f[n_] := Block[{a = {1}, i, k}, Do[k = 1; While[Or[Mod[k, a[[i - 1]]] != a[[i - 1]] - 1, MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 120 (* Michael De Vlieger, Jun 11 2015 *)
A[n_]:= If[n<4, n, If[EvenQ[n], 2A[n/2] + 1, 2A[(n - 1)/2]]]; Table[A[n], {n, 100}] (* Indranil Ghosh, Mar 21 2017 *)
f[lst_List] := Block[{k = 2, m = lst[[-1]]}, While[ MemberQ[lst, k] || 1 + Mod[k, m] != m, k++]; Append[lst, k]]; Nest[f, {1}, 70] (* Robert G. Wilson v, Jan 22 2018 *)

A(n) = if(n<4, n, if(n%2, 2*A(n\2), 2*A(n/2)+1));
for(n=1, 50, print1(A(n), ", ")) \\ Indranil Ghosh, Mar 21 2017

a <- 1:3 # If it were c(1, 3, 2), it would be A054429
maxn <- 50 # by choice
#
for(n in 2:maxn){
  a[2*n  ] <- 2*a[n]+1
  a[2*n+1] <- 2*a[n]
}
#
a
# Yosu Yurramendi, Jun 08 2015

For n >= 2: If a(n-1) = 2^m, m=positive integer, then a(n)= 2^(m+1)-1. If a(n-1) = 3*2^m, m= nonnegative integer, then a(n) = 3*2^(m+1)-1. Otherwise, a(n) = a(n-1) -1.

For n >= 2: a(2*n) = 2*a(n)+1, a(2*n+1) = 2*a(n). - Yosu Yurramendi, Jun 08 2015

More terms from Klaus Brockhaus

A284459 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A245327/A245328, and A162911/A162912 (Drib) into A020651/A020650 (Yu-Ting inverted).

Original entry on oeis.org

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

Offset: 1

Yosu Yurramendi, Mar 27 2017

nonn

The inverse permutation is A284460.

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

Cf. A002487, A020650, A020651, A162911, A162912, A245327, A245328, A284460.

maxrow <- 12 # by choice
a <- 1
b01 <- 1
for(m in 0:maxrow){
  b01 <- c(b01, c(1-b01[2^m:(2^(m+1)-1)], b01[2^m:(2^(m+1)-1)]) )
  for(k in 0:(2^m-1)){
    a[2^(m+1) +       k] <- a[2^m + k] + 2^(m + b01[2^(m+1) +       k])
    a[2^(m+1) + 2^m + k] <- a[2^m + k] + 2^(m + b01[2^(m+1) + 2^m + k])
}}
a
# Yosu Yurramendi, Mar 27 2017

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
for(i in 2:(length(anbit) - 1))
   anbit[i] <- 1 - bitwXor(anbit[i], anbit[i-1])
a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Apr 25 2021

a(n) = A258996(A231551(n)) = A231551(A092569(n)), n > 0 . - Yosu Yurramendi, Apr 10 2017

A231551 Position of n in A231550.

Original entry on oeis.org

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

Offset: 0

Alex Ratushnyak, Nov 10 2013

This permutation transforms the enumeration system of positive irreducible fractions A002487/A002487' (Calkin-Wilf) into the enumeration system A020651/A020650, and A162911/A162912 (Drib) the enumeration system into A245327/A245326. - Yosu Yurramendi, Jun 16 2015

Cf. A003188, A006068, A231550.

Join[{0, 1}, Table[d = Reverse@IntegerDigits[n, 2]; FromDigits[Reverse@Append[FoldList[BitXor, d[[1]], Most@Rest@d], d[[-1]]], 2], {n, 2, 67}]] (* Ivan Neretin, Dec 28 2016 *)

for n in range(99):
  bits = [0]*64
  orig = [0]*64
  l = int.bit_length(int(n))
  t = n
  for i in range(l):
    bits[i] = orig[i] = t&1
    t>>=1
  #for i in range(1, l-1):  bits[i] ^= orig[i-1]   # A231550
  for i in range(1, l-1):  bits[i] ^= bits[i-1]   # A231551
  #for i in range(l-1):  bits[i] ^= orig[i+1]      # A003188
  #for i in range(1, l):  bits[l-1-i] ^= bits[l-i]  # A006068
  t = 0
  for i in range(l):  t += bits[i]<

R

maxrow <- 8 # by choice b01 <- 0 # b01 is going to be A010059 a <- 1 for(m in 0:maxrow) for(k in 0:(2^m-1)){ b01[2^(m+1)+ k] <- b01[2^m+k] a[2^(m+1)+ k] <- a[2^m+k] + 2^(m+b01[2^(m+1)+ k]) b01[2^(m+1)+2^m+k] <- 1 - b01[2^m+k] a[2^(m+1)+2^m+k] <- a[2^m+k] + 2^(m+b01[2^(m+1)+2^m+k]) } (a <- c(0,a)) # Yosu Yurramendi, Apr 10 2017

R

maxblock <- 8 # 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 for(i in 2:(length(anbit) - 1)) anbit[i] <- bitwXor(anbit[i], anbit[i-1]) # ?bitwXor a <- c(a, sum(anbit*2^(0:(length(anbit) - 1)))) } (a <- c(0,a)) # Yosu Yurramendi, Apr 25 2021

Formula

A231550(a(n)) = a(A231550(n)) = n.

a(n) = A258996(A284460(n)) = A284459(A092569(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

a(n) = A054429(A153154(n)), n > 0. - Yosu Yurramendi, Oct 04 2021

cp's OEIS Frontend

A092570 Primes p which become a prime q under transformation of inner bits of binary representation in A092569. In binary representation of p, transformation of inner bits, 1 <-> 0, gives binary representation of q.

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

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A054429 Simple self-inverse permutation of natural numbers: List each block of 2^n numbers (from 2^n to 2^(n+1) - 1) in reverse order.

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A065190 Self-inverse permutation of the positive integers: 1 is fixed, followed by an infinite number of adjacent transpositions (n n+1).

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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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A258746 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A007305/A047679 (Stern-Brocot) into the enumeration system A162909/A162910 (Bird), and vice versa.

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A117120 a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).

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