This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A253563 #33 Feb 15 2021 04:42:28
%S A253563 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,
%T A253563 75,21,125,35,49,11,64,48,72,40,108,60,100,28,162,90,150,42,250,70,98,
%U A253563 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
%N A253563 Permutation of natural numbers: a(0) = 1, a(1) = 2; after which, a(2n) = A253560(a(n)), a(2n+1) = A253550(a(n)).
%C A253563 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:
%C A253563 1
%C A253563 |
%C A253563 ...................2...................
%C A253563 4 3
%C A253563 8......../ \........6 9......../ \........5
%C A253563 / \ / \ / \ / \
%C A253563 / \ / \ / \ / \
%C A253563 / \ / \ / \ / \
%C A253563 16 12 18 10 27 15 25 7
%C A253563 32 24 36 20 54 30 50 14 81 45 75 21 125 35 49 11
%C A253563 etc.
%C A253563 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.
%C A253563 Note that the indexing of sequence starts from 0, although its range starts from one.
%C A253563 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
%H A253563 Antti Karttunen, <a href="/A253563/b253563.txt">Table of n, a(n) for n = 0..8191</a>
%H A253563 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A253563 a(0) = 1, a(1) = 2; after which, a(2n) = A253560(a(n)), a(2n+1) = A253550(a(n)).
%F A253563 As a composition of other permutations:
%F A253563 a(n) = A122111(A005940(n+1)).
%F A253563 a(n) = A253565(A054429(n)).
%F A253563 Other identities and observations. For all n >= 0:
%F A253563 A002110(n) = a(A002450(n)). [Primorials occur at positions (4^n - 1)/3.]
%F A253563 For all n >= 1: a(2n) - a(2n+1) > 0. [See the comment above.]
%p A253563 a:= proc(n) local m; m:= n; [0]; while m>0 do `if`(1=
%p A253563 irem(m, 2, 'm'), map(x-> x+1, %), [%[], 0]) od:
%p A253563 `if`(n=0, 1, mul(ithprime(i), i=%))
%p A253563 end:
%p A253563 seq(a(n), n=0..100); # _Alois P. Heinz_, Aug 23 2017
%t A253563 p[n_] := p[n] = FactorInteger[n][[-1, 1]];
%t A253563 b[n_] := n p[n];
%t A253563 c[1] = 1; c[n_] := (n/p[n]) NextPrime[p[n]];
%t A253563 a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], c[a[(n-1)/2]]];
%t A253563 a /@ Range[0, 100] (* _Jean-François Alcover_, Feb 15 2021 *)
%o A253563 (Scheme, two versions, the other one using memoizing definec-macro)
%o A253563 (definec (A253563 n) (cond ((< n 2) (+ 1 n)) ((even? n) (A253560 (A253563 (/ n 2)))) (else (A253550 (A253563 (/ (- n 1) 2))))))
%o A253563 (define (A253563 n) (A122111 (A005940 (+ 1 n))))
%Y A253563 Inverse: A253564.
%Y A253563 Cf. A252737 (row sums), A252738 (row products).
%Y A253563 Cf. A002110, A002450, A005940, A122111, A252464, A253561, A253550, A253560, A253565, A054429, A252464.
%K A253563 nonn,look,tabf
%O A253563 0,2
%A A253563 _Antti Karttunen_, Jan 03 2015