A254115 -id:A254115 - cp's OEIS frontend

A254099 Fixed points of A254115 and A254116.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 16, 20, 22, 24, 28, 32, 40, 44, 48, 56, 64, 80, 88, 96, 112, 128, 160, 176, 192, 224, 256, 317, 320, 352, 384, 448, 512, 634, 640, 704, 768, 896, 1024, 1268, 1280, 1408, 1536, 1792, 2048, 2536, 2560, 2816, 3072, 3584, 4096, 5072, 5120, 5632, 6144, 7168, 8192, 10144, 10240, 11264

Offset: 1

Antti Karttunen, Feb 04 2015

nonn

If n is a member, then 2n is as well, and vice versa, thus the sequence is completely determined by its odd terms: 1, 3, 5, 7, 11, 317, 15989189, 16964283, ...

Antti Karttunen, Table of n, a(n) for n = 1..152

Subsequence: A000079.

Cf. A254115, A254116.

\\ Needs also code from A254116:
isA254099(n) = (A254116(n) == n);
i=0; for(n=1, 123456789, if(isA254099(n), i++; write("b254099.txt", i, " ", n)))
(Scheme, with Antti Karttunen's IntSeq-library)
(define A254099 (FIXED-POINTS 1 1 A254115))

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.]

A254117 Permutation of natural numbers: a(n) = A254104(A249746(1+n)-1).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 10, 7, 11, 21, 9, 31, 16, 15, 23, 27, 63, 13, 43, 8, 19, 53, 14, 37, 45, 47, 95, 26, 22, 17, 511, 127, 12, 91, 39, 18, 85, 62, 29, 2047, 30, 38, 111, 20, 54, 75, 42, 55, 157, 35, 46, 174, 25, 79, 24, 59, 77, 303, 123, 94, 78, 167, 87, 223, 61, 103, 699, 191, 41, 107, 183, 125, 431, 28, 71, 84, 60, 93, 110, 767, 51

Offset: 1

Antti Karttunen, Feb 05 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: A254118.

Other related permutations: A249746, A254104, A254115.

from sympy import factorint, nextprime
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a254104(n):
    if n==0: return 0
    if n%3==0: return 1 + 2*a254104(2*n/3 - 1)
    elif n%3==1: return 1 + 2*a254104(2*(n - 1)/3)
    else: return 2*a254104((n - 2)/3 + 1)
def a254115(n): return a254104(a048673(n))
def a(n): return (a254115(2*n + 1) - 1)/2 # Indranil Ghosh, Jun 06 2017

a(n) = A254104(A249746(1+n)-1).

a(n) = (A254115((2*n)+1)-1) / 2. [Obtained also from the odd bisection of A254115.]

cp's OEIS Frontend

A254099 Fixed points of A254115 and A254116.

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

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A254117 Permutation of natural numbers: a(n) = A254104(A249746(1+n)-1).

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