A264985 -id:A264985 - cp's OEIS frontend

A265354 Permutation of nonnegative integers: a(n) = A263273(A264985(n)).

0, 1, 3, 2, 4, 9, 6, 10, 12, 7, 5, 19, 8, 13, 27, 18, 28, 36, 21, 11, 57, 24, 37, 30, 15, 31, 39, 22, 16, 64, 23, 14, 55, 20, 46, 58, 25, 17, 73, 26, 40, 81, 54, 82, 108, 63, 29, 171, 72, 109, 90, 45, 85, 117, 66, 34, 192, 69, 38, 165, 60, 100, 174, 75, 35, 219, 78, 118, 84, 33, 91, 93, 48, 32, 138, 51, 112, 111, 42, 94, 120, 67

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

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

Inverse: A265353.
Cf. A263273, A264985.
Cf. also A265352, A265355, A265356.

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 a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(n)) # Indranil Ghosh, May 22 2017

(define (A265354 n) (A263273 (A264985 n)))

a(n) = A263273(A264985(n)).

A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 5, 7, 19, 8, 13, 31, 24, 28, 37, 15, 11, 33, 18, 27, 30, 21, 36, 39, 14, 16, 46, 23, 25, 73, 69, 55, 64, 17, 22, 58, 26, 40, 94, 78, 85, 112, 51, 34, 100, 72, 82, 91, 75, 109, 118, 42, 32, 96, 20, 35, 105, 60, 99, 102, 45, 29, 87, 54, 81, 84, 57, 90, 93, 48, 38, 114, 63, 108, 111, 66, 117, 120, 41

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

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

Inverse: A265354.
Cf. A263273, A264985.
Cf. also A265341, A265351, A265355, A265356.

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 a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a264985(a263273(n)) # Indranil Ghosh, May 22 2017

(define (A265353 n) (A264985 (A263273 n)))

a(n) = A264985(A263273(n)).

A265355 Permutation of nonnegative integers: a(n) = A263272(A264985(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 6, 10, 12, 5, 11, 7, 8, 13, 27, 18, 28, 36, 15, 29, 21, 24, 37, 30, 33, 31, 39, 14, 32, 16, 17, 38, 19, 20, 34, 22, 23, 35, 25, 26, 40, 81, 54, 82, 108, 45, 83, 63, 72, 109, 90, 99, 85, 117, 42, 86, 48, 51, 110, 57, 60, 88, 66, 69, 89, 75, 78, 118, 84, 87, 91, 93, 96, 92, 102, 105, 112, 111, 114, 94, 120, 41

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of permutations obtained from the bisections of A263273.

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

Inverse: A265356.
Cf. A263272, A263273, A264985.
Cf. also A265351, A265352, A265353, A265354, A265357, A265358.

(define (A265355 n) (A263272 (A264985 n)))

a(n) = A263272(A264985(n)).

A265356 Permutation of nonnegative integers: a(n) = A264985(A263272(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 6, 11, 12, 5, 7, 10, 8, 13, 27, 18, 29, 30, 15, 32, 33, 20, 35, 36, 21, 38, 39, 14, 16, 19, 23, 25, 28, 24, 34, 37, 17, 22, 31, 26, 40, 81, 54, 83, 84, 45, 86, 87, 56, 89, 90, 57, 92, 93, 42, 95, 96, 59, 98, 99, 60, 101, 102, 47, 104, 105, 62, 107, 108, 63, 110, 111, 48, 113, 114, 65, 116, 117, 66, 119, 120, 41

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of permutations obtained from the bisections of A263273.

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

Inverse: A265355.
Cf. A263272, A263273, A264985.
Cf. also A265351, A265352, A265353, A265354, A265357, A265358.

(define (A265356 n) (A264985 (A263272 n)))

a(n) = A264985(A263272(n)).

A264991 Permutation of nonnegative integers: a(n) = A264989(A264985(n)).

Original entry on oeis.org

0, 1, 5, 2, 4, 14, 6, 16, 17, 3, 7, 8, 11, 13, 41, 15, 43, 44, 18, 19, 59, 56, 49, 50, 47, 52, 53, 9, 10, 32, 29, 22, 23, 20, 25, 26, 12, 34, 35, 38, 40, 122, 42, 124, 125, 45, 46, 140, 137, 130, 131, 128, 133, 134, 54, 55, 167, 164, 58, 176, 60, 178, 179, 57, 169, 170, 173, 148, 149, 51, 151, 152, 48

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

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

Inverse: A264992.
Cf. A263273, A264985, A264989.
Cf. also A264975, A264976.

(define (A264991 n) (A264989 (A264985 n)))

a(n) = A264989(A264985(n)).

A264992 Permutation of nonnegative integers: a(n) = A264985(A264989(n)).

Original entry on oeis.org

0, 1, 3, 9, 4, 2, 6, 10, 11, 27, 28, 12, 36, 13, 5, 15, 7, 8, 18, 19, 33, 99, 31, 32, 96, 34, 35, 81, 82, 30, 90, 85, 29, 87, 37, 38, 108, 109, 39, 117, 40, 14, 42, 16, 17, 45, 46, 24, 72, 22, 23, 69, 25, 26, 54, 55, 21, 63, 58, 20, 60, 100, 101, 297, 298, 102, 306, 94, 95, 285, 97, 98, 288

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

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

Inverse: A264991.
Cf. A263273, A264985, A264989.
Cf. also A264975, A264976.

(define (A264992 n) (A264985 (A264989 n)))

a(n) = A264985(A264989(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)).

A264996 Self-inverse permutation of natural numbers: a(n) = (1/2) * (1+A263273(2n -1)) = 1 + A264985(n-1).

Original entry on oeis.org

1, 2, 4, 3, 5, 10, 7, 11, 13, 6, 8, 12, 9, 14, 28, 19, 29, 37, 16, 20, 34, 25, 32, 31, 22, 38, 40, 15, 17, 33, 24, 23, 30, 21, 35, 39, 18, 26, 36, 27, 41, 82, 55, 83, 109, 46, 56, 100, 73, 86, 91, 64, 110, 118, 43, 47, 97, 70, 59, 88, 61, 101, 115, 52, 74, 106, 79, 95, 85, 58, 92, 112, 49, 65, 103, 76

Offset: 1

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 1..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985.

Cf. also A266401, A266403, A266415, A266416.

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]]]; Array[(1/2) (1 + f[2 # - 1]) &, {76}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

a(n) = (1/2) * (1+A263273((2*n)-1)).

a(n) = 1 + A264985(n-1).

A266189 Self-inverse permutation of nonnegative integers: a(n) = A263273(A264985(A263273(n))).

Original entry on oeis.org

0, 1, 3, 2, 4, 10, 6, 9, 12, 7, 5, 11, 8, 13, 37, 24, 28, 31, 21, 19, 57, 18, 27, 30, 15, 36, 39, 22, 16, 34, 23, 17, 35, 69, 29, 32, 25, 14, 38, 26, 40, 118, 78, 109, 112, 75, 46, 100, 72, 82, 91, 51, 85, 94, 66, 64, 192, 20, 73, 219, 60, 171, 138, 63, 55, 165, 54, 81, 84, 33, 90, 111, 48, 58, 174, 45, 108, 93, 42, 117, 120, 67, 49

Offset: 0

Antti Karttunen, Jan 02 2016

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. also A265902, A266190.

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[f@ t[[f@ n + 1]], {n, 0, 82}] (* 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 a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(a263273(n))) # Indranil Ghosh, May 22 2017

(define (A266189 n) (A263273 (A264985 (A263273 n))))

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

A263273 Bijective base-3 reverse: a(0) = 0; for n >= 1, a(n) = A030102(A038502(n)) * A038500(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 19, 12, 13, 22, 21, 16, 25, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 55, 30, 37, 64, 57, 46, 73, 36, 31, 58, 39, 40, 67, 66, 49, 76, 63, 34, 61, 48, 43, 70, 75, 52, 79, 54, 29, 56, 33, 38, 65, 60, 47, 74, 45, 32, 59, 42, 41, 68, 69, 50, 77, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Here the base-3 reverse has been adjusted so that the maximal suffix of trailing zeros (in base-3 representation A007089) stays where it is at the right side, and only the section from the most significant digit to the least significant nonzero digit is reversed, thus making this sequence a self-inverse permutation of nonnegative integers.
Because successive powers of 3 and 9 modulo 2, 4 and 8 are always either constant 1, 1, 1, ... or alternating 1, -1, 1, -1, ... it implies similar simple divisibility rules for 2, 4 and 8 in base 3 as e.g. 3, 9 and 11 have in decimal base (see the Wikipedia-link). As these rules do not depend on which direction they are applied from, it means that this bijection preserves the fact whether a number is divisible by 2, 4 or 8, or whether it is not. Thus natural numbers are divided to several subsets, each of which is closed with respect to this bijection. See the Crossrefs section for permutations obtained from these sections.
When polynomials over GF(3) are encoded as natural numbers (coefficients presented with the digits of the base-3 expansion of n), this bijection works as a multiplicative automorphism of the ring GF(3)[X]. This follows from the fact that as there are no carries involved, the multiplication (and thus also the division) of such polynomials could be as well performed by temporarily reversing all factors (like they were seen through mirror). This implies also that the sequences A207669 and A207670 are closed with respect to this bijection.

			For n = 15, A007089(15) = 120. Reversing this so that the trailing zero stays at the right yields 210 = A007089(21), thus a(15) = 21 and vice versa, a(21) = 15.

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

Bisections: A264983, A264984.
Cf. A000035, A007089, A010873, A030102, A038500, A038502, A207669, A207670.
Cf. also A057889, A264965, A264966, A264980.
Permutations induced by various sections: A263272 (a(2n)/2), A264974 (a(4n)/4), A264978 (a(8n)/8), A264985, A264989.
Cf. also A004488, A140263, A140264, A246207, A246208 (other base-3 related permutations).

r[n_] := FromDigits[Reverse[IntegerDigits[n, 3]], 3]; b[n_] := n/ 3^IntegerExponent[n, 3]; c[n_] := n/b[n]; a[0]=0; a[n_] := r[b[n]]*c[n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 29 2015 *)

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 a(n): return 0 if n==0 else a030102(a038502(n))*a038500(n) # Indranil Ghosh, May 22 2017

(define (A263273 n) (if (zero? n) n (* (A030102 (A038502 n)) (A038500 n))))

a(0) = 0; for n >= 1, a(n) = A030102(A038502(n)) * A038500(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.]
A010873(a(n)) = 0 if and only if A010873(n) = 0. [See the comments section.]

cp's OEIS Frontend

A265354 Permutation of nonnegative integers: a(n) = A263273(A264985(n)).

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A265353 Permutation of nonnegative integers: a(n) = A264985(A263273(n)).

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A265355 Permutation of nonnegative integers: a(n) = A263272(A264985(n)).

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A265356 Permutation of nonnegative integers: a(n) = A264985(A263272(n)).

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A264991 Permutation of nonnegative integers: a(n) = A264989(A264985(n)).

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A264992 Permutation of nonnegative integers: a(n) = A264985(A264989(n)).

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

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A264996 Self-inverse permutation of natural numbers: a(n) = (1/2) * (1+A263273(2n -1)) = 1 + A264985(n-1).

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