A246200 -id:A246200 - cp's OEIS frontend

A264967 Permutation of nonnegative integers: a(n) = A263272(A246200(n)).

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 7, 12, 29, 14, 15, 32, 23, 18, 13, 20, 33, 38, 27, 24, 25, 22, 17, 28, 39, 30, 37, 16, 21, 34, 35, 36, 41, 26, 19, 40, 31, 42, 95, 68, 105, 54, 47, 96, 119, 98, 69, 116, 55, 86, 83, 56, 71, 78, 59, 60, 107, 74, 99, 92, 65, 114, 113, 44, 81, 50, 87, 72, 89, 82, 45, 104, 53, 110, 101, 80, 51

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A264968.
Cf. A246200, A263272.
Cf. also A264965, A264966.

(define (A264967 n) (A263272 (A246200 n)))

a(n) = A263272(A246200(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of numbers.]

A264968 Permutation of nonnegative integers: a(n) = A246200(A263272(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 7, 12, 19, 14, 15, 32, 27, 18, 39, 20, 33, 26, 17, 24, 25, 38, 23, 28, 13, 30, 41, 16, 21, 34, 35, 36, 31, 22, 29, 40, 37, 42, 123, 68, 75, 86, 47, 96, 135, 70, 81, 152, 77, 46, 53, 56, 107, 110, 59, 60, 163, 82, 99, 108, 65, 142, 111, 44, 51, 134, 57, 72, 139, 62, 147, 156, 83, 58, 87, 80, 69

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A264967.
Cf. A246200, A263272.
Cf. also A264965, A264966.

(define (A264968 n) (A246200 (A263272 n)))

a(n) = A246200(A263272(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of numbers.]

A280509 a(n) = A051064(A246200(n)); 3-adic valuation of A057889(3*n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..10921
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 73.
Index entries for sequences related to binary expansion of n

Cf. A007949, A057889, A246200, A280508.
Differs from A051064 for the first time at n=23, where a(23) = 4, while A051064(23) = 1.
Cf. also A265331.

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)));
A280509(n) = valuation(A057889(3*n),3); \\ Antti Karttunen, Dec 26 2024

(define (A280509 n) (A051064 (A246200 n)))

a(n) = A007949(A057889(3*n)).

a(n) = A051064(A246200(n)).

A286588 a(n) = A278222(A246200(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 6, 6, 12, 4, 12, 8, 16, 2, 6, 6, 12, 6, 30, 12, 36, 4, 12, 12, 24, 8, 24, 16, 32, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 36, 72, 4, 12, 12, 36, 12, 60, 24, 72, 8, 24, 24, 48, 16, 48, 32, 64, 2, 6, 6, 12, 6, 30, 12, 24, 6, 30, 30, 60, 12, 60, 24, 48, 6, 30, 30, 60, 30, 210, 60, 180, 12, 60, 60, 180, 36

Offset: 0

Antti Karttunen, Jun 03 2017

Antti Karttunen, Table of n, a(n) for n = 0..10921
Indranil Ghosh, Python Program to generate the sequence
Index entries for sequences related to binary expansion of n

Cf. A246200, A278222, A286589 (rgs-version of this sequence).

(define (A286588 n) (A278222 (A246200 n)))

a(n) = A278222(A246200(n)).

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.

Original entry on oeis.org

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

A263272 Self-inverse permutation of nonnegative integers: a(n) = A263273(2*n) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 7, 12, 13, 14, 15, 32, 23, 18, 29, 20, 33, 38, 17, 24, 35, 26, 27, 28, 19, 30, 37, 16, 21, 34, 25, 36, 31, 22, 39, 40, 41, 42, 95, 68, 45, 86, 59, 96, 113, 50, 69, 104, 77, 54, 83, 56, 87, 110, 47, 60, 101, 74, 99, 92, 65, 114, 119, 44, 51, 98, 71, 72, 89, 62, 105, 116, 53, 78, 107, 80, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263273.
Bisections: A264986, A264987.
Cf. also A246200, A264967, A264968, A264974, A264978, A264975, A264976, A264984.

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]]]; Table[f[2 n]/2, {n, 0, 81}] (* 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 a(n): return a263273(2*n)/2 # Indranil Ghosh, May 23 2017

(define (A263272 n) (/ (A263273 (+ n n)) 2))

a(n) = A263273(2*n) / 2 = A264984(n) / 2.
As a composition of related permutations:
a(n) = A264974(A264975(n)) = A264976(A264974(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.]
A264974(n) = a(2n)/2. [Thus the restriction onto even numbers induces yet another permutation.]

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)).

A264993 Self-inverse permutation of nonnegative integers: a(n) = A264994(3*n)/3.

Original entry on oeis.org

0, 1, 3, 2, 4, 5, 11, 7, 12, 19, 15, 6, 8, 18, 14, 10, 16, 17, 13, 9, 20, 21, 43, 27, 44, 75, 59, 23, 28, 71, 55, 39, 48, 67, 51, 35, 76, 83, 47, 31, 60, 79, 63, 22, 24, 70, 54, 38, 32, 66, 50, 34, 72, 82, 46, 30, 56, 78, 62, 26, 40, 74, 58, 42, 64, 65, 49, 33, 68, 81, 45, 29, 52, 77, 61, 25, 36, 73, 57, 41, 80, 69

Offset: 0

Antti Karttunen, Dec 07 2015

Antti Karttunen, Table of n, a(n) for n = 0..21845
Index entries for sequences that are permutations of the natural numbers

Cf. A264994.

Cf. also A246200, A265335.

(define (A264993 n) (/ (A264994 (* 3 n)) 3))

cp's OEIS Frontend

A264967 Permutation of nonnegative integers: a(n) = A263272(A246200(n)).

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A264968 Permutation of nonnegative integers: a(n) = A246200(A263272(n)).

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A280509 a(n) = A051064(A246200(n)); 3-adic valuation of A057889(3*n).

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A286588 a(n) = A278222(A246200(n)).

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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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A263272 Self-inverse permutation of nonnegative integers: a(n) = A263273(2*n) / 2.

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

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

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