A057889 -id:A057889 - cp's OEIS frontend

A366275 The Cat's tongue permutation: a(n) = A163511(A057889(n)).

1, 2, 4, 3, 8, 9, 6, 5, 16, 27, 18, 15, 12, 25, 10, 7, 32, 81, 54, 45, 36, 75, 30, 21, 24, 125, 50, 35, 20, 49, 14, 11, 64, 243, 162, 135, 108, 225, 90, 63, 72, 375, 150, 105, 60, 147, 42, 33, 48, 625, 250, 175, 100, 245, 70, 55, 40, 343, 98, 77, 28, 121, 22, 13, 128, 729, 486, 405, 324, 675, 270, 189, 216, 1125

Offset: 0

Antti Karttunen, Oct 06 2023

"Cat's tongue" refers to the look of the scatter plot of this sequence.

Cf. A000040, A000225, A007814, A057889, A163511, A209229, A290251, A366276 (inverse map), A366277 (fixed points of map n -> a(n)), A366278, A366279, A366280, A366281 [= A052409(a(n))], A366282 [= a(n)-n], A366283 [= gcd(n,a(n))].

Cf. also A163511, A253563, A366263 (compare the scatter plots).

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)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366275(n) = A163511(A057889(n));

from sympy import prime
def A366275(n):
    if n:
        k, c, m = int(bin(n>>(r:=(~n & n-1).bit_length()))[:1:-1],2)<>= s+1
        return m*prime(c)
    return 1 # Chai Wah Wu, Oct 08 2023

For n >= 0, A001222(a(n)) = A290251(n).
For n >= 1, A007814(a(n)) = A135523(n) = A007814(n) + A209229(n). [Like A163511, also this permutation preserves the 2-adic valuation of n, except when n is a power of two, in which cases that value is incremented by one.]
For n >= 1, a(2*n) = 2*a(n).
For n >= 1, a(A000225(n)) = A000040(n).

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.]

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.]

A246200 Self-inverse permutation of natural numbers: a(n) = A057889(3*n) / 3.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 27 2014

In binary system, 3 ("11" in binary), has a similar shortcut rule for divisibility as eleven has in decimal system. This rule doesn't depend on which end of the number representation it is applied from, thus, if we reverse the number 3*n with "balanced bit-reverse" (A057889), the result should still be divisible by 3. Moreover, because the reversing operation is itself a self-inverse involution, and the prime factorization of any natural number is unique, we get a self-inverse permutation of nonnegative integers when we divide the bit-reversed result with 3.

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

Cf. A036215, A057889, A003714, A048724, A083822, A083824.

def a057889(n):
    x=bin(n)[2:]
    y=x[::-1]
    return int(str(int(y))+(len(x) - len(str(int(y))))*'0', 2)
def a(n): return a057889(3*n)//3
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

(define (A246200 n) (/ (A057889 (* 3 n)) 3))

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

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.]

A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 1, 14, 15, 16, 17, 18, 1, 20, 21, 2, 3, 24, 1, 2, 27, 28, 3, 30, 31, 32, 33, 34, 7, 36, 1, 2, 5, 40, 1, 42, 3, 4, 45, 6, 1, 48, 7, 2, 51, 4, 3, 54, 1, 56, 5, 6, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 1, 14, 3, 72, 73, 2, 15, 4, 3, 10, 7, 80, 1, 2, 9, 84, 85, 6, 1, 8, 3, 90, 1, 12, 93, 2, 5, 96, 1, 14, 99, 4, 9, 102, 1, 8, 15, 6

Offset: 1

Antti Karttunen, Jan 09 2017

a(n) = the maximal GF(2)[X]-divisor of n which in base 2 is either a palindrome or becomes a palindrome if trailing 0's are omitted.
More precisely: a(n) = the unique term m of A057890 for which A280500(n,m) > 0 and A091222(m) >= A091222(k) for all such terms k of A057890 for which A280500(n,k) > 0.
All terms are in A057890 and each term of A057890 occurs an infinite number of times.

Cf. A048720, A057889, A057890, A091222, A091255, A280501, A280503, A280506.

(define (A280505 n) (A091255bi n (A057889 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n,A057889(n)).
Other identities. For all n >= 1:
a(A057889(n)) = a(n).
A048720(a(n), A280506(n)) = n.

A266351 Start with a(1) = 1, then always choose for a(n) the least unused number such that A057889(a(n)a(n-1)) = A057889(a(n)) A057889(a(n-1)), where A057889 is a bijective base-2 reverse.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 11, 17, 13, 32, 15, 24, 18, 20, 19, 33, 21, 36, 28, 27, 56, 34, 22, 64, 23, 65, 25, 40, 35, 72, 42, 48, 30, 51, 60, 66, 26, 68, 37, 96, 31, 99, 62, 128, 29, 129, 38, 80, 49, 73, 70, 130, 39, 256, 41, 131, 74, 136, 44, 132, 46, 257, 43, 258, 45, 260, 47, 512, 50, 133, 76, 160, 67, 84, 97, 137, 112, 54

Offset: 1

Antti Karttunen, Dec 28 2015

Equally: always choose for a(n) the least unused number such that a(n)*a(n-1) = A057889(A057889(a(n)) * A057889(a(n-1))).
Note that the adjacent terms of permutation A266195 satisfy the same condition, except that permutation is not the lexicographically earliest sequence of this kind (because it has a more restrictive condition). See A266194.
This is a bijection for the same reason that A266195 is. Any high enough 2^k will always save the permutation of being stuck, and will also immediately pick up as its succeeding pair the least term unused so far.

Antti Karttunen, Table of n, a(n) for n = 1..1450
Index entries for sequences that are permutations of the natural numbers

Inverse: A266352.
Cf. A057889, A266194.
Cf. A266195, A265405, A266405 (similar sequences).

A280508 a(n) = n XOR A057889(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 10, 0, 0, 12, 10, 0, 10, 12, 0, 0, 10, 0, 0, 0, 0, 0, 18, 0, 12, 20, 30, 0, 12, 0, 30, 24, 0, 20, 18, 0, 18, 20, 0, 24, 30, 0, 12, 0, 30, 20, 12, 0, 18, 0, 0, 0, 0, 0, 34, 0, 20, 36, 54, 0, 0, 24, 34, 40, 20, 60, 54, 0, 20, 24, 54, 0, 0, 60, 34, 48, 20, 0, 54, 40, 0, 36, 34, 0, 34, 36, 0, 40, 54, 0, 20, 48

Offset: 0

Antti Karttunen, Jan 09 2017

Cf. A003987, A057889, A280505, A280506, A280507, A280509.

Cf. A057890 (positions of zeros).

(define (A280508 n) (A003987bi n (A057889 n))) ;; Where A003987bi implements A003987 (bitwise-XOR).

a(n) = A003987(n,A057889(n)) = n XOR A057889(n).
Other identities. For all n >= 0:
a(A057889(n)) = a(n).

A345201 Bit-reverse the odd part of the Zeckendorf representation of n: a(n) = A022290(A057889(A003714(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 18, 17, 20, 21, 22, 23, 24, 30, 26, 27, 31, 29, 25, 28, 32, 33, 34, 35, 36, 37, 48, 39, 43, 49, 42, 40, 44, 50, 53, 47, 38, 41, 45, 51, 52, 46, 54, 55, 56, 57, 58, 77, 60, 69, 78, 63, 64, 70, 79, 85

Offset: 0

Rémy Sigrist, Jun 10 2021

This sequence is a self-inverse permutation of the nonnegative integers.

This sequence is similar to A343150 and to A344682.

Rémy Sigrist, Table of n, a(n) for n = 0..10946
Rémy Sigrist, PARI program for A345201
Rémy Sigrist, Alternate PARI program for A345201
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A022290, A057889, A343150, A344682.

PARI
```
See Links section.
```

a(n) < A000045(k) for any n < A000045(k).

cp's OEIS Frontend

A366275 The Cat's tongue permutation: a(n) = A163511(A057889(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A246200 Self-inverse permutation of natural numbers: a(n) = A057889(3*n) / 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A266351 Start with a(1) = 1, then always choose for a(n) the least unused number such that A057889(a(n)*a(n-1)) = A057889(a(n)) * A057889(a(n-1)), where A057889 is a bijective base-2 reverse.

Views

Author

Keywords

Comments

Links

Crossrefs

A266351 Start with a(1) = 1, then always choose for a(n) the least unused number such that A057889(a(n)a(n-1)) = A057889(a(n)) A057889(a(n-1)), where A057889 is a bijective base-2 reverse.