A263273 -id:A263273 - cp's OEIS frontend

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

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.]

A264974 Self-inverse permutation of natural numbers: a(n) = A263273(4*n) / 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 19, 12, 13, 14, 15, 8, 17, 18, 11, 20, 21, 34, 43, 48, 25, 52, 27, 28, 55, 30, 37, 46, 57, 22, 49, 36, 31, 58, 39, 40, 41, 42, 23, 50, 45, 32, 59, 24, 35, 44, 51, 26, 53, 54, 29, 56, 33, 38, 47, 60, 61, 142, 63, 88, 169, 102, 115, 124, 129, 70, 151, 144, 97, 178, 75, 106, 133, 156, 79, 160, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..4921
Index entries for sequences that are permutations of the natural numbers

Terms of A264986 halved.
Cf. A263272, A263273, A264978.
Cf. also A264975, A264976.

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(4*n)/4 # Indranil Ghosh, May 25 2017

a(n) = A263273(4*n) / 4.
a(n) = A264986(n) / 2 = A263272(2*n) / 2.
As a composition of related permutations:
a(n) = A264975(A263272(n)) = A263272(A264976(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.]
A264978(n) = a(2n)/2. [Thus the restriction onto even numbers induces yet another permutation.]

A264965 Permutation of nonnegative integers: a(n) = A263273(A057889(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 05 2015

Perform an adjusted reverse of n in base 2, followed by another adjusted reverse in base 3. "Adjusted reverse" here means the digit-reversing operation in which the tail of trailing zeros (in the base in question) is fixed, while the portion from the most significant digit to the least significant nonzero digit is reversed.

What percentage of the cycles are finite? (See the scatter-plot and A264969, also A264972, A264973.)

Antti Karttunen, Table of n, a(n) for n = 0..10000
Indranil Ghosh, Python program to generate the sequence

Inverse: A264966.
Cf. A000035, A057889, A263273, A264969, A264972, A264973.
Cf. also A264967, A264968.

(define (A264965 n) (A263273 (A057889 n)))

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

A264966 Permutation of nonnegative integers: a(n) = A057889(A263273(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 25, 12, 11, 26, 21, 16, 19, 18, 13, 20, 15, 14, 29, 24, 17, 22, 27, 28, 59, 30, 41, 64, 39, 58, 73, 36, 31, 46, 57, 40, 97, 66, 35, 100, 63, 34, 47, 48, 53, 98, 105, 44, 121, 54, 23, 56, 33, 50, 65, 60, 61, 82, 45, 32, 55, 42, 37, 68, 81, 38, 89, 72, 49, 62, 51, 52, 113, 114, 43, 80, 69

Offset: 0

Antti Karttunen, Dec 05 2015

Perform an adjusted reverse of n in base 3, followed by another adjusted reverse in base 2. "Adjusted reverse" here means a digit-reversing operation where the suffix of trailing zeros (in the base in question) stays as it is at the right side, and only the section from the most significant digit to the least significant nonzero digit is reversed.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Inverse: A264965.
Cf. A000035, A057889, A263273.
Cf. also A264967, A264968.

(define (A264966 n) (A057889 (A263273 n)))

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

A265352 Permutation of nonnegative integers: a(n) = A263273(A263272(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 19, 8, 9, 10, 5, 12, 13, 22, 21, 64, 23, 18, 55, 20, 57, 58, 25, 24, 73, 26, 27, 28, 11, 30, 31, 16, 15, 46, 17, 36, 37, 14, 39, 40, 67, 66, 199, 68, 63, 190, 65, 192, 193, 70, 69, 208, 71, 54, 163, 56, 165, 166, 61, 60, 181, 62, 171, 172, 59, 174, 175, 76, 75, 226, 77, 72, 217, 74, 219, 220, 79, 78, 235, 80, 81

Offset: 0

Antti Karttunen, Dec 07 2015

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

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

Inverse: A265351.
Cf. A263272, A263273, A264974, A265368.
Cf. also A265354, 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 a(n): return a263273(a263273(2*n)/2) # Indranil Ghosh, Jun 08 2017

(define (A265352 n) (A263273 (A263272 n)))

a(n) = A263273(A263272(n)).
As a composition of other related permutations:
a(n) = A265368(A264974(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).

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.]

A265351 Permutation of nonnegative integers: a(n) = A263272(A263273(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Dec 07 2015

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

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

Inverse: A265352.
Cf. A263272, A263273, A264974, A265367.
Cf. also A265342, A265353, 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 a263272(n): return a263273(2*n)/2
def a(n): return a263272(a263273(n)) # Indranil Ghosh, May 25 2017

(define (A265351 n) (A263272 (A263273 n)))

a(n) = A263272(A263273(n)).
As a composition of other related permutations:
a(n) = A264974(A265367(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
a(n) = A265342(n)/2.

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

Original entry on oeis.org

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)).

A265369 Self-inverse permutation of nonnegative integers: a(n) = A057889(A263273(A057889(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 11, 12, 25, 26, 21, 16, 19, 18, 17, 20, 15, 22, 59, 24, 13, 14, 27, 28, 29, 30, 41, 64, 39, 58, 53, 36, 97, 98, 33, 40, 31, 66, 121, 44, 63, 50, 61, 48, 73, 46, 105, 100, 35, 54, 65, 56, 57, 34, 23, 60, 47, 82, 45, 32, 55, 42, 137, 68, 69, 142, 131, 72, 49, 74, 219, 76, 155, 234, 79, 80, 81, 62, 173

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..19684
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. also A265329, A265904, A266190, A266401, A266403, 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[n_] := FromDigits[Reverse@ IntegerDigits[n, 2], 2] 2^IntegerExponent[n, 2]; Prepend[#, 0] &@ Table[g@ f@ g@ n, {n, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 and Ivan Neretin at A057889 *)

(define (A265369 n) (A057889 (A263273 (A057889 n))))

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

cp's OEIS Frontend

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

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A264974 Self-inverse permutation of natural numbers: a(n) = A263273(4*n) / 4.

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A264965 Permutation of nonnegative integers: a(n) = A263273(A057889(n)).

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A264966 Permutation of nonnegative integers: a(n) = A057889(A263273(n)).

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A265352 Permutation of nonnegative integers: a(n) = A263273(A263272(n)).

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

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A265351 Permutation of nonnegative integers: a(n) = A263272(A263273(n)).

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

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