A030102 -id:A030102 - cp's OEIS frontend

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

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)).

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)).

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.]

A321464 Reverse nonzero digits in ternary expansion of n and convert back to decimal.

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, 31, 58, 57, 34, 61, 36, 37, 64, 39, 40, 67, 66, 49, 76, 63, 46, 73, 48, 43, 70, 75, 52, 79, 54, 29, 56, 33, 32, 59, 60, 35, 62, 45, 38, 65, 42, 41

Offset: 0

Rémy Sigrist, Nov 10 2018

This sequence is a self-inverse permutation of nonnegative integers with fixed points A321473.
See A321474 for the decimal variant.
The binary variant simply corresponds to the identity (A001477).

			The first values at prime indices, alongside the corresponding ternary expansions, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   2     2       2          2
   3     3      10         10
   5     7      12         21
   7     5      21         12
  11    19     102        201
  13    13     111        111
  17    25     122        221
  19    11     201        102
  23    23     212        212
  29    55    1002       2001
  31    31    1011       1011
  37    37    1101       1101
  41    67    1112       2111
  43    49    1121       1211

Cf. A001477, A004488, A030102, A321473, A321474.

a[n_] := Block[{x = IntegerDigits[n, 3], t}, t = Flatten@ Position[x, 1 | 2]; x[[Reverse@ t]] = x[[t]]; FromDigits[x, 3]]; Array[a, 68, 0] (* Giovanni Resta, Sep 17 2019 *)

a(n,base=3) = my (d=digits(n,base),t=Vecrev(select(sign,d)),i=0); for (j=1, #d, if (d[j], d[j] = t[i++])); fromdigits(d,base)

a(3 * n) = 3 * a(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.]

A264984 Even bisection of A263273; terms of A263262 doubled.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 22, 16, 18, 20, 14, 24, 26, 28, 30, 64, 46, 36, 58, 40, 66, 76, 34, 48, 70, 52, 54, 56, 38, 60, 74, 32, 42, 68, 50, 72, 62, 44, 78, 80, 82, 84, 190, 136, 90, 172, 118, 192, 226, 100, 138, 208, 154, 108, 166, 112, 174, 220, 94, 120, 202, 148, 198, 184, 130

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A000035, A010673, A010873, A263272, A263273, A264983, A264975.

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) # Indranil Ghosh, May 22 2017

Scheme
```
(define (A264984 n) (A263273 (+ n n)))
```

a(n) = 2 * A263272(n).
a(n) = A263273(2*n).
Other identities. For all n >= 0:
A010873(a(n)) = 2 * A000035(n) = A010673(n).

A265342 Permutation of even numbers: a(n) = 2 * A265351(n).

Original entry on oeis.org

0, 2, 4, 6, 8, 22, 12, 10, 16, 18, 20, 58, 24, 26, 76, 66, 64, 70, 36, 14, 40, 30, 28, 34, 48, 46, 52, 54, 56, 166, 60, 62, 184, 174, 172, 178, 72, 74, 220, 78, 80, 238, 228, 226, 232, 198, 68, 202, 192, 190, 196, 210, 208, 214, 108, 38, 112, 42, 44, 130, 120, 118, 124, 90, 32, 94, 84, 82, 88, 102, 100, 106, 144

Offset: 0

Antti Karttunen, Dec 07 2015

nonn

Iterating this sequence as 1, a(1), a(a(1)), a(a(a(1))), ... yields A264980.

Antti Karttunen, Table of n, a(n) for n = 0..9841

Cf. A265351.

Cf. also A265341, A263273, A264980.

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 2*a263272(a263273(n)) # Indranil Ghosh, May 25 2017

Scheme
```
(define (A265342 n) (* 2 (A265351 n)))
```

a(n) = 2 * A265351(n).

A264983 Odd bisection of A263273.

Original entry on oeis.org

1, 3, 7, 5, 9, 19, 13, 21, 25, 11, 15, 23, 17, 27, 55, 37, 57, 73, 31, 39, 67, 49, 63, 61, 43, 75, 79, 29, 33, 65, 47, 45, 59, 41, 69, 77, 35, 51, 71, 53, 81, 163, 109, 165, 217, 91, 111, 199, 145, 171, 181, 127, 219, 235, 85, 93, 193, 139, 117, 175, 121, 201, 229, 103, 147, 211

Offset: 0

Antti Karttunen, Dec 05 2015

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A263273, A264984, A264985.

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 = Select[f /@ Range@ 130, OddQ] (* 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 + 1) # Indranil Ghosh, May 22 2017

(define (A264983 n) (A263273 (+ 1 n n)))

a(n) = A263273(2n + 1).

A264986 Even bisection of A263272; terms of A264974 doubled.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 32, 18, 20, 38, 24, 26, 28, 30, 16, 34, 36, 22, 40, 42, 68, 86, 96, 50, 104, 54, 56, 110, 60, 74, 92, 114, 44, 98, 72, 62, 116, 78, 80, 82, 84, 46, 100, 90, 64, 118, 48, 70, 88, 102, 52, 106, 108, 58, 112, 66, 76, 94, 120, 122, 284, 126, 176, 338, 204, 230, 248, 258, 140, 302, 288

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Cf. A263272, A264974, A264987.

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)/2 # Indranil Ghosh, May 23 2017

Scheme
```
(define (A264986 n) (A263272 (+ n n)))
```

a(n) = A263272(2*n).
a(n) = 2 * A264974(n).
a(n) = A263273(4*n)/2.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A321464 Reverse nonzero digits in ternary expansion of n and convert back to decimal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Scheme

Formula

A264984 Even bisection of A263273; terms of A263262 doubled.

Views

Author

Keywords

Crossrefs

Programs

Python

Scheme

Formula

A265342 Permutation of even numbers: a(n) = 2 * A265351(n).