A265369 -id:A265369 - cp's OEIS frontend

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 49, 36, 41, 50, 57, 40, 37, 42, 53, 52, 45, 58, 61, 48, 35, 38, 51, 44, 43, 54, 59, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 97, 68, 81, 98, 113

Offset: 0

Marc LeBrun, Sep 25 2000

The original name was "Bit-reverse of n, including as many leading as trailing zeros." - Antti Karttunen, Dec 25 2024
A permutation of integers consisting only of fixed points and pairs. a(n)=n when n is a binary palindrome (including as many leading as trailing zeros), otherwise a(n)=A003010(n) (i.e. n has no axis of symmetry). A057890 gives the palindromes (fixed points, akin to A006995) while A057891 gives the "antidromes" (pairs). See also A280505.
This is multiplicative in domain GF(2)[X], i.e. with carryless binary arithmetic. A193231 is another such permutation of natural numbers. - Antti Karttunen, Dec 25 2024

			a(6)=6 because 0110 is a palindrome, but a(11)=13 because 1011 reverses into 1101.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384, May 30 2016 [First 8192 terms from _Ivan Neretin_, Jul 09 2015]
Index entries for sequences related to binary expansion of n
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Cf. A030101, A000265, A006519, A006995, A057890, A057891, A280505, A280508, A331166 [= min(n,a(n))], A366378 [k for which a(k) = k (mod 3)], A369044 [= A014963(a(n))].
Similar permutations for other bases: A263273 (base-3), A264994 (base-4), A264995 (base-5), A264979  (base-9).
Other related (binary) permutations: A056539, A193231.
Compositions of this permutation with other binary (or other base-related) permutations: A264965, A264966, A265329, A265369, A379471, A379472.
Compositions with permutations involving prime factorization: A245450, A245453, A266402, A266404, A293448, A366275, A366276.
Other derived permutations: A246200 [= a(3*n)/3], A266351, A302027, A302028, A345201, A356331, A356332, A356759, A366389.
See also A235027 (which is not a permutation).

Table[FromDigits[Reverse[IntegerDigits[n, 2]], 2]*2^IntegerExponent[n, 2], {n, 71}] (* Ivan Neretin, Jul 09 2015 *)

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2))); \\ Antti Karttunen, Dec 25 2024

def a(n):
    x = bin(n)[2:]
    y = x[::-1]
    return int(str(int(y))+(len(x) - len(str(int(y))))*'0', 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

def A057889(n): return int(bin(n>>(m:=(~n&n-1).bit_length()))[-1:1:-1],2)<Chai Wah Wu, Dec 25 2024

a(n) = A030101(A000265(n)) * A006519(n), with a(0)=0.

Clarified the name with May 30 2016 comment from N. J. A. Sloane, and moved the old name to the comments - Antti Karttunen, Dec 25 2024

A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 6, 13, 10, 11, 20, 9, 14, 71, 22, 7, 26, 19, 12, 23, 16, 21, 24, 41, 18, 53, 28, 31, 56, 29, 38, 107, 58, 67, 74, 61, 32, 197, 40, 25, 68, 59, 50, 137, 64, 73, 62, 49, 44, 227, 76, 27, 80, 55, 30, 89, 34, 43, 66, 37, 48, 91, 46, 69, 60, 35, 42, 65, 70, 15, 78, 47, 36, 119, 52

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6560
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A250469, A250470, A263273.
Cf. A265369, A265904, A266190, A266401 (other conjugates or similar derivations of A263273).
Cf. also A250471, A250472, A264996, A266404, A266415, A266416, A266645, A266646.

(define (A266403 n) (A250470 (A263273 (A250469 n))))

a(n) = A250470(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266415(A266645(n)) = A266646(A266416(n)).
a(n) = A250472(A264996(A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A265329 Self-inverse permutation of nonnegative integers: a(n) = A263273(A057889(A263273(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 17, 12, 19, 26, 15, 16, 11, 18, 13, 20, 21, 22, 55, 24, 25, 14, 27, 28, 65, 30, 67, 32, 39, 38, 35, 36, 37, 34, 33, 40, 145, 42, 73, 100, 45, 46, 61, 48, 79, 226, 219, 76, 121, 54, 23, 56, 57, 70, 59, 60, 47, 82, 63, 64, 29, 66, 31, 68, 81, 58, 217, 72, 43, 74, 75, 52, 193, 174, 49, 80, 69, 62, 221

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..16384
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A057889, A263273, A264965, A264966.
Cf. A266402, A266404.
Cf. also A265369.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; g[n_] := FromDigits[Reverse@ IntegerDigits[n, 2], 2] 2^IntegerExponent[n, 2]; Prepend[#, 0] &@ Table[f@ g@ f@ n, {n, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 and Ivan Neretin at A057889 *)

(define (A265329 n) (A263273 (A057889 (A263273 n))))

a(n) = A263273(A057889(A263273(n))).
As a composition of related permutations:
a(n) = A264965(A263273(n)).
a(n) = A263273(A264966(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266401 Self-inverse permutation of natural numbers: a(n) = A064989(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 17, 8, 13, 6, 11, 20, 9, 34, 71, 16, 7, 26, 19, 12, 23, 22, 21, 40, 41, 18, 227, 68, 31, 142, 29, 32, 53, 14, 67, 52, 61, 38, 107, 24, 25, 46, 59, 44, 65, 42, 73, 80, 49, 82, 197, 36, 33, 454, 55, 136, 137, 62, 43, 284, 37, 58, 571, 64, 45, 106, 35, 28, 89, 134, 15, 104, 47

Offset: 1

Antti Karttunen, Jan 02 2016

Shift primes in the prime factorization of n one step towards larger primes (A003961), then apply the bijective base-3 reverse (A263273) to the resulting odd number, which yields another (or same) odd number, then shift primes in the prime factorization of that second odd number one step back towards smaller primes (A064989).

Antti Karttunen, Table of n, a(n) for n = 1..6560
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A003961, A064989, A263273.
Cf. A265369, A265904, A266190, A266403 (other conjugates or similar sequences derived from A263273).
Cf. also A048673, A064216, A264996, A266402, A266407, A266408.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; g[p_?PrimeQ] := g[p] = Prime[PrimePi@ p + 1]; g[1] = 1; g[n_] := g[n] = Times @@ (g[First@ #]^Last@ # &) /@ FactorInteger@ n; h[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Table[h@ f@ g@ n, {n, 82}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 and A263273 *)

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A266401 = n -> A064989(A263273(A003961(n)));
for(n=1, 6560, write("b266401.txt", n, " ", A266401(n)));

(define (A266401 n) (A064989 (A263273 (A003961 n))))

a(n) = A064989(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A064216(A264996(A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 6, 29, 8, 9, 10, 5, 12, 13, 38, 33, 92, 17, 18, 83, 20, 87, 110, 35, 24, 89, 26, 27, 28, 7, 30, 37, 32, 15, 86, 23, 36, 31, 14, 39, 40, 119, 114, 281, 44, 99, 254, 65, 276, 335, 98, 51, 260, 71, 54, 245, 56, 249, 326, 101, 60, 263, 74, 261, 272, 47, 330, 353, 116, 105, 278, 53, 72, 251, 62, 267, 332, 107, 78, 269, 80, 81, 82, 19

Offset: 0

Antti Karttunen, Jan 02 2016

A263273 conjugated with the permutation obtained from its even bisection.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263272, A263273, A265351, A265352.
Cf. also A265902.
Cf. A265369, A266190, A266401, A266403 (other conjugates or similar derivations of A263273).

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@g[n] h[n]]]; t = Table[f[2 n]/2, {n, 0, 1000}]; Table[t[[f[t[[n + 1]]] + 1]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return a263272(a263273(a263272(n))) # Indranil Ghosh, May 25 2017

(define (A265904 n) (A263272 (A263273 (A263272 n))))

a(n) = A263272(A263273(A263272(n))).
As a composition of related permutations:
a(n) = A263272(A265352(n)).
a(n) = A265351(A263272(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266190 Self-inverse permutation of nonnegative integers: a(n) = A264985(A263273(A264985(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 19, 12, 13, 14, 15, 16, 17, 24, 11, 69, 21, 25, 23, 18, 22, 26, 31, 28, 73, 30, 27, 46, 33, 55, 58, 37, 36, 64, 39, 40, 41, 42, 43, 44, 51, 32, 150, 48, 52, 50, 45, 49, 53, 78, 34, 213, 57, 35, 204, 60, 61, 231, 75, 38, 210, 66, 79, 68, 20, 70, 77, 72, 29, 207, 63, 76, 71, 54, 67, 80, 94, 85, 235

Offset: 0

Antti Karttunen, Jan 02 2016

A263273 conjugated with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985, A265353, A265354.
Cf. A265369, A265904, A266401, A266403 (other conjugates or similar derivations of A263273).
Cf. also A266189.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; s = Select[f /@ Range@ 5000, OddQ]; t = Table[(s[[n + 1]] - 1)/2, {n, 0, 1000}]; Table[t[[f[t[[n + 1]]] + 1]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

(define (A266190 n) (A264985 (A263273 (A264985 n))))

a(n) = A264985(A263273(A264985(n))).
As a composition of related permutations:
a(n) = A265353(A264985(n)).
a(n) = A264985(A265354(n)).

A266641 Permutation of nonnegative integers: a(n) = A264965(2*n) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 13, 12, 7, 14, 15, 32, 23, 18, 35, 20, 33, 26, 19, 24, 29, 38, 27, 28, 17, 30, 37, 16, 21, 34, 113, 36, 41, 50, 87, 40, 31, 42, 77, 104, 45, 110, 101, 96, 25, 22, 69, 68, 95, 54, 47, 56, 39, 86, 83, 60, 59, 74, 99, 92, 65, 114, 97, 44, 81, 70, 49, 72, 89, 82, 75, 88, 73, 66, 121, 80, 51

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences that are permutations of the natural numbers

Inverse: A266642.

Cf. A057889, A263272, A264965, A265329, A265369, A266635, A266636, A266643, A266644.

a(n) = A264965(2*n) / 2.
As a composition of related permutations:
a(n) = A263272(A057889(n)).

A266642 Permutation of nonnegative integers: a(n) = A264966(2*n) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 13, 8, 9, 10, 7, 12, 11, 14, 15, 32, 29, 18, 23, 20, 33, 50, 17, 24, 49, 22, 27, 28, 25, 30, 41, 16, 21, 34, 19, 36, 31, 26, 57, 40, 37, 42, 125, 68, 45, 106, 55, 96, 71, 38, 81, 88, 89, 54, 101, 56, 117, 118, 61, 60, 83, 82, 99, 116, 65, 78, 119, 52, 51, 70, 113, 72, 77, 62, 75, 92, 43, 114, 107, 80, 69

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..9842
Index entries for sequences that are permutations of the natural numbers

Inverse: A266641.

Cf. A057889, A263272, A264966, A265329, A265369, A266635, A266636, A266643, A266644.

(define (A266642 n) (/ (A264966 (* 2 n)) 2))

a(n) = A264966(2*n) / 2.
As a composition of related permutations:
a(n) = A057889(A263272(n)).

A266643 Permutation of nonnegative integers: a(n) = A264965(3*n) / 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 19, 12, 11, 22, 21, 16, 25, 18, 13, 20, 15, 14, 27, 24, 73, 58, 23, 28, 39, 30, 37, 64, 57, 46, 17, 36, 67, 26, 55, 40, 31, 66, 49, 76, 51, 54, 65, 48, 41, 50, 75, 44, 163, 34, 29, 56, 77, 78, 61, 60, 53, 74, 45, 32, 59, 42, 43, 68, 81, 70, 33, 72, 35, 82, 63, 52, 71, 38, 47, 80, 69

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..10921
Index entries for sequences that are permutations of the natural numbers

Inverse: A266644.

Cf. A246200, A263273, A264965, A265329, A265369, A266635, A266636, A266641, A266642.

a(n) = A264965(3*n) / 3.
As a composition of related permutations:
a(n) = A263273(A246200(n)).

A266644 Permutation of nonnegative integers: a(n) = A264966(3*n) / 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 13, 12, 19, 22, 21, 16, 35, 18, 11, 20, 15, 14, 27, 24, 17, 38, 23, 28, 55, 30, 41, 64, 71, 54, 73, 36, 31, 78, 29, 40, 49, 66, 67, 52, 63, 34, 79, 48, 43, 50, 45, 76, 61, 46, 39, 56, 33, 26, 65, 60, 59, 82, 75, 32, 47, 42, 37, 68, 81, 70, 77, 72, 25, 62, 51, 44, 57, 58, 83, 80, 69

Offset: 0

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 0..19682
Index entries for sequences that are permutations of the natural numbers

Inverse: A266643.
Cf. A246200, A263273, A264966, A265329, A265369, A266635, A266636, A266641, A266642.
Differs from A264965 for the first time at n=17, where a(17) = 35, while A264965(17) = 25.

(define (A266644 n) (/ (A264966 (* 3 n)) 3))

a(n) = A264966(3*n) / 3.
As a composition of related permutations:
a(n) = A246200(A263273(n)).

cp's OEIS Frontend

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

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A266403 Self-inverse permutation of natural numbers: a(n) = A250470(A263273(A250469(n))).

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A265329 Self-inverse permutation of nonnegative integers: a(n) = A263273(A057889(A263273(n))).

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A266401 Self-inverse permutation of natural numbers: a(n) = A064989(A263273(A003961(n))).

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A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

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A266190 Self-inverse permutation of nonnegative integers: a(n) = A264985(A263273(A264985(n))).

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A266641 Permutation of nonnegative integers: a(n) = A264965(2*n) / 2.

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A266642 Permutation of nonnegative integers: a(n) = A264966(2*n) / 2.

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