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A253563 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253560(a(n)), a(2n+1) = A253550(a(n)). 16
1, 2, 4, 3, 8, 6, 9, 5, 16, 12, 18, 10, 27, 15, 25, 7, 32, 24, 36, 20, 54, 30, 50, 14, 81, 45, 75, 21, 125, 35, 49, 11, 64, 48, 72, 40, 108, 60, 100, 28, 162, 90, 150, 42, 250, 70, 98, 22, 243, 135, 225, 63, 375, 105, 147, 33, 625, 175, 245, 55, 343, 77, 121, 13, 128, 96, 144, 80, 216, 120, 200, 56, 324, 180, 300, 84, 500, 140, 196, 44 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A253560 to the parent, and each child to the right is obtained by applying A253550 to the parent:

                                     1

                                     |

                  ...................2...................

                 4                                       3

       8......../ \........6                   9......../ \........5

      / \                 / \                 / \                 / \

     /   \               /   \               /   \               /   \

    /     \             /     \             /     \             /     \

  16       12         18       10         27       15         25       7

32  24   36  20     54  30   50  14     81  45   75  21    125  35   49 11

etc.

Sequence A253565 is the mirror image of the same tree. Also in binary trees A005940 and A163511 the terms on level of the tree are some permutation of the terms present on the level n of this tree. A252464(n) tells distance of n from 1 in all these trees. Of these four trees, this is the one where the left child is always larger than the right child.

Note that the indexing of sequence starts from 0, although its range starts from one.

a(n) (n>=1) can be obtained by the composition of a bijection between {1,2,3,4,...} and the set of integer partitions and a bijection between the set of integer partitions and {2,3,4,...}. Explanation on the example n=10. Write 2*n = 20 as a binary number: 10100. Consider a Ferrers board whose southeast border is obtained by replacing each 1 by an east step and each 0 by a north step. We obtain the Ferrers board of the partition p = (2,2,1). Finally, a(10) = 2'*2'*1', where m' = m-th prime. Thus, a(10)= 3*3*2 = 18. - Emeric Deutsch, Sep 17 2016

LINKS

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

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(0) = 1, a(1) = 2; after which, a(2n) = A253560(a(n)), a(2n+1) = A253550(a(n)).

As a composition of other permutations:

a(n) = A122111(A005940(n+1)).

a(n) = A253565(A054429(n)).

Other identities and observations. For all n >= 0:

A002110(n) =  a(A002450(n)). [Primorials occur at positions (4^n - 1)/3.]

For all n >= 1: a(2n) - a(2n+1) > 0. [See the comment above.]

MAPLE

a:= proc(n) local m; m:= n; [0]; while m>0 do `if`(1=

      irem(m, 2, 'm'), map(x-> x+1, %), [%[], 0]) od:

      `if`(n=0, 1, mul(ithprime(i), i=%))

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2017

PROG

(Scheme, two versions, the other one using memoizing definec-macro)

(definec (A253563 n) (cond ((< n 2) (+ 1 n)) ((even? n) (A253560 (A253563 (/ n 2)))) (else (A253550 (A253563 (/ (- n 1) 2))))))

(define (A253563 n) (A122111 (A005940 (+ 1 n))))

CROSSREFS

Inverse: A253564.

Cf. A252737 (row sums), A252738 (row products).

Cf. A002110, A002450, A005940, A122111, A252464, A253561, A253550, A253560, A253565, A054429, A252464.

Sequence in context: A209636 A243491 A271863 * A294044 A243072 A243346

Adjacent sequences:  A253560 A253561 A253562 * A253564 A253565 A253566

KEYWORD

nonn,look,tabf

AUTHOR

Antti Karttunen, Jan 03 2015

STATUS

approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)