A254103 -id:A254103 - cp's OEIS frontend

A291760 Binary encoding of 2-digits in ternary representation of A254103(n).

0, 0, 1, 0, 1, 0, 3, 2, 1, 0, 1, 2, 5, 0, 3, 0, 1, 4, 7, 2, 1, 4, 5, 0, 9, 0, 1, 4, 5, 0, 1, 2, 1, 8, 9, 2, 13, 0, 3, 6, 1, 4, 7, 0, 9, 0, 3, 4, 17, 4, 7, 8, 1, 8, 11, 2, 9, 12, 15, 2, 1, 4, 5, 6, 1, 20, 23, 2, 17, 4, 5, 6, 25, 0, 1, 10, 5, 8, 11, 0, 1, 8, 9, 12, 13, 0, 1, 2, 17, 0, 1, 4, 5, 8, 9, 0, 33, 8, 9, 12, 13, 16, 17, 0, 1, 16, 17, 2, 21, 0, 3, 6, 17

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A254103, A286632, A286633, A289814, A291759, A291763.

a(n) = A289814(A254103(n)).

A304740 Restricted growth sequence transform of A304760(n), formed from 1-digits in ternary representation of A254103(n).

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 1, 5, 6, 6, 2, 3, 7, 6, 8, 9, 4, 1, 6, 10, 1, 1, 5, 5, 11, 11, 3, 10, 12, 12, 8, 13, 6, 6, 14, 3, 15, 11, 1, 16, 17, 12, 14, 3, 17, 12, 2, 9, 18, 19, 2, 20, 4, 1, 12, 16, 4, 1, 11, 21, 12, 12, 2, 22, 4, 1, 23, 10, 19, 19, 14, 5, 24, 24, 2, 20, 7, 6, 25, 26, 23, 23, 3, 21, 19, 19, 25, 5, 23, 23, 10, 21, 1, 1, 10, 13, 27, 27, 21, 28, 1, 1

Offset: 0

Antti Karttunen, May 29 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A254103, A289813, A304760.

Cf. also A304746.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304760(n) = A289813(A254103(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304740 = rgs_transform(vector(65538,n,A304760(n-1)));
A304740(n) = v304740[1+n];

A304746 Restricted growth sequence transform of A291760(n), formed from 2-digits in ternary representation of A254103(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 2, 4, 5, 1, 3, 1, 2, 6, 7, 4, 2, 6, 5, 1, 8, 1, 2, 6, 5, 1, 2, 4, 2, 9, 8, 4, 10, 1, 3, 11, 2, 6, 7, 1, 8, 1, 3, 6, 12, 6, 7, 9, 2, 9, 13, 4, 8, 14, 15, 4, 2, 6, 5, 11, 2, 16, 17, 4, 12, 6, 5, 11, 18, 1, 2, 19, 5, 9, 13, 1, 2, 9, 8, 14, 10, 1, 2, 4, 12, 1, 2, 6, 5, 9, 8, 1, 20, 9, 8, 14, 10, 21, 12, 1, 2, 21, 12

Offset: 0

Antti Karttunen, May 29 2018

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A254103, A289814, A291760.

Cf. also A304740, A304748.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291760(n) = A289814(A254103(n));
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v304746 = rgs_transform(vector(65537,n,A291760(n-1)));
A304746(n) = v304746[1+n];

A292241 The 3-adic valuation of A254103(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 4, 0, 0, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Sep 12 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

One less than the even bisection of A292242.
Cf. A007814, A007949, A254103, A292240.
Cf. also A292251, A292261.

a(n) = A007814(1+A292240(n)).
a(n) = A007949(A254103(n)).
For n >= 1, a(n) = A007949(3*A254103(n)) - 1.
For n >= 1, a(n) = A292242(2n)-1.

A304760 Binary encoding of 1-digits in ternary representation of A254103(n).

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 0, 0, 6, 4, 4, 1, 2, 7, 4, 5, 14, 3, 0, 4, 10, 0, 0, 6, 6, 12, 12, 2, 10, 8, 8, 5, 30, 4, 4, 9, 2, 15, 12, 0, 22, 11, 8, 9, 2, 11, 8, 1, 14, 19, 16, 1, 26, 3, 0, 8, 22, 3, 0, 12, 18, 8, 8, 1, 62, 3, 0, 20, 10, 16, 16, 9, 6, 28, 28, 1, 26, 7, 4, 13, 46, 20, 20, 2, 18, 16, 16, 13, 6, 20, 20, 10, 18

Offset: 0

Antti Karttunen, May 29 2018

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A254103, A289813, A304740 (rgs-transform).

Cf. also A291760.

A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304760(n) = A289813(A254103(n));

a(n) = A289813(A254103(n)).

A254116 Permutation of natural numbers: a(n) = A064216(A254103(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 13, 10, 11, 12, 9, 14, 17, 16, 41, 26, 23, 20, 15, 22, 19, 24, 67, 18, 37, 28, 47, 34, 29, 32, 27, 82, 61, 52, 73, 46, 43, 40, 89, 30, 21, 44, 59, 38, 31, 48, 111, 134, 107, 36, 57, 74, 33, 56, 149, 94, 79, 68, 83, 58, 25, 64, 359, 54, 181, 164, 193, 122, 101, 104, 229, 146, 49, 92, 85, 86, 71, 80, 185, 178, 139, 60, 95, 42, 39

Offset: 1

Antti Karttunen, Feb 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A254115.
Fixed points: A254099.
Related permutations: A064216, A254103, A254118.

default(primelimit, 2^30);
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)};
A064216(n) = A064989((2*n)-1);
A254103(n) = { if(0==n,0,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2)); };
A254116(n) = A064216(A254103(n));
for(n=1, 8192, write("b254116.txt", n, " ", A254116(n)));

from sympy import factorint, prevprime, floor
from operator import mul
def a064216(n):
    f=factorint(2*n - 1)
    return 1 if n==1 else reduce(mul, [prevprime(i)**f[i] for i in f])
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n/2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))/2)
def a(n): return a064216(a254103(n)) # Indranil Ghosh, Jun 06 2017

(define (A254116 n) (A064216 (A254103 n)))

a(n) = A064216(A254103(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [Even bisection halved gives back the sequence itself.]
A254118(n) = (a((2*n)+1) - 1)/2. [Likewise, the odd bisection induces A254118.]

A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 4, 1, 3, 3, 5, 3, 5, 2, 5, 4, 6, 3, 4, 2, 4, 2, 6, 2, 4, 3, 6, 1, 3, 4, 6, 3, 5, 4, 7, 4, 6, 4, 5, 5, 7, 2, 5, 3, 5, 3, 7, 5, 7, 3, 5, 4, 6, 3, 7, 6, 8, 4, 4, 3, 5, 5, 7, 6, 8, 4, 6, 3, 5, 6, 8, 3, 5, 5, 7, 5, 7, 3, 6, 4, 6, 5, 8, 1, 3, 5, 6, 2, 4, 4, 6, 2, 4, 2, 8, 4, 6, 6, 8, 2, 4, 2, 6, 3, 5, 4, 7, 4, 6, 5, 8, 5, 7, 5, 9, 5, 7, 4, 5

Offset: 0

Antti Karttunen, Jun 03 2017

Reflecting the structure of A254103 also this sequence can be represented as a binary tree:
                                     0
                                     |
                  ...................1...................
                 2                                       1
       3......../ \........2                   4......../ \........2
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   4       1           3       3           5       3           5       2
  5 4     6 3         4 2     4 2         6 2     4 3         6 1     3 4
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A053735, A254103, A286585, A286633.

from sympy.ntheory.factor_ import digits
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n/2) - 1
    else: return floor((3*(1 + a254103((n - 1)/2)))/2)
def a(n): return sum(digits(a254103(n), 3)[1:]) # Indranil Ghosh, Jun 06 2017

(define (A286632 n) (A053735 (A254103 n)))

a(n) = A053735(A254103(n)).
a(n) = A056239(A286633(n)).
For all n >= 0, a(A000079(n)) = n+1.

A286633 Base-3 {digit+1} product of A254103: a(n) = A006047(A254103(n)).

Original entry on oeis.org

1, 2, 3, 2, 6, 4, 9, 3, 12, 2, 6, 6, 18, 8, 18, 4, 24, 12, 27, 6, 12, 3, 9, 4, 36, 4, 12, 6, 36, 2, 6, 12, 48, 6, 18, 12, 54, 16, 36, 9, 24, 24, 54, 4, 18, 8, 18, 6, 72, 24, 54, 6, 24, 12, 27, 6, 72, 36, 81, 12, 12, 6, 18, 18, 96, 36, 81, 12, 36, 6, 18, 36, 108, 8, 24, 18, 72, 24, 54, 8, 48, 12, 36, 18, 108, 2, 6, 24, 36, 4, 12, 12, 36, 3, 9, 4

Offset: 0

Antti Karttunen, Jun 03 2017

Reflecting the structure of A254103 also this sequence can be represented as a binary tree:
                                     1
                                     |
                  ...................2...................
                 3                                       2
       6......../ \........4                   9......../ \........3
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  12       2           6       6          18       8          18       4
24  12   27 6        12 3     9 4       36  4    12 6       36  2     6 12
etc.

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A000079, A006047, A042950, A254103, A286586, A286587, A286632.

from sympy.ntheory.factor_ import digits
from operator import mul
from functools import reduce
def a006047(n):
    d=digits(n, 3)
    return reduce(mul, [1 + d[i] for i in range(1, len(d))])
def a254103(n):
    if n==0: return 0
    if n%2==0: return 3*a254103(n//2) - 1
    else: return (3*(1 + a254103((n - 1)//2)))//2
def a(n): return a006047(a254103(n)) # Indranil Ghosh, Jun 06 2017

(define (A286633 n) (A006047 (A254103 n)))

a(n) = A006047(A254103(n)).

For n >= 0, a(A000079(n)) = A042950(n).

A292240 Binary encoding of 0-digits in ternary representation of A254103(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 4, 3, 2, 1, 0, 3, 2, 1, 0, 7, 6, 0, 0, 3, 2, 4, 0, 0, 0, 1, 8, 0, 0, 6, 4, 4, 4, 2, 0, 8, 8, 6, 4, 4, 4, 5, 0, 0, 0, 1, 12, 3, 2, 0, 0, 8, 8, 9, 4, 11, 10, 0, 0, 3, 2, 4, 0, 0, 0, 2, 16, 3, 2, 1, 0, 15, 14, 0, 8, 11, 10, 1, 8, 7, 6, 5, 0, 19, 18, 1, 16, 15, 14, 13, 8, 11

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A254103, A291760, A291770, A292241, A292242.

Cf. also A292250, A292260.

a(n) = A291770(A254103(n)).

A292242 Number of trailing 2-digits in ternary representation of A254103(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1

Offset: 0

Antti Karttunen, Sep 12 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A007814, A007949, A254103, A291760, A292241 (even bisection subtracted by one).

Cf. also A292252, A292262.

(define (A292242 n) (A007949 (+ 1 (A254103 n))))
(define (A292242 n) (A007814 (+ 1 (A291760 n))))
(define (A292242 n) (cond ((zero? n) n) ((odd? n) 0) (else (+ 1 (A292241 (/ n 2))))))

a(n) = A007949(1+A254103(n)).
a(n) = A007814(1+A291760(n)).
a(0) = 0, after which, a(2n) = 1 + A292241(n/2), a(2n+1) = 0.

cp's OEIS Frontend

A291760 Binary encoding of 2-digits in ternary representation of A254103(n).

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A304740 Restricted growth sequence transform of A304760(n), formed from 1-digits in ternary representation of A254103(n).

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A304746 Restricted growth sequence transform of A291760(n), formed from 2-digits in ternary representation of A254103(n).

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A292241 The 3-adic valuation of A254103(n).

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A304760 Binary encoding of 1-digits in ternary representation of A254103(n).

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A254116 Permutation of natural numbers: a(n) = A064216(A254103(n)).

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A286632 Base-3 digit sum of A254103: a(n) = A053735(A254103(n)).

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A286633 Base-3 {digit+1} product of A254103: a(n) = A006047(A254103(n)).

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A292240 Binary encoding of 0-digits in ternary representation of A254103(n).

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