A253563 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253560(a(n)), a(2n+1) = A253550(a(n)).

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

Offset: 0

Author

Antti Karttunen, Jan 03 2015

Keywords

• nonn
• look
• tabf

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

Crossrefs

Inverse: A253564.

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

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

Programs

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

Mathematica

p[n_] := p[n] = FactorInteger[n][[-1, 1]];
b[n_] := n p[n];
c[1] = 1; c[n_] := (n/p[n]) NextPrime[p[n]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], c[a[(n-1)/2]]];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 15 2021 *)

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.]