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a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).
11

%I #10 Jun 11 2020 23:44:04

%S 1,2,4,3,16,8,9,6,256,32,64,12,81,18,36,5,65536,512,1024,48,4096,128,

%T 144,24,6561,162,324,27,1296,72,25,10,4294967296,131072,262144,768,

%U 1048576,2048,2304,96,16777216,8192,16384,192,20736,288,576,20,43046721,13122,26244,243,104976,648,729,54,1679616,2592,5184,108,625,50,100,15

%N a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

%C This irregular table can be represented as a binary tree. Each child to the left is obtained by squaring the parent, and each child to the right is obtained by applying A334747 to the parent:

%C 1

%C |

%C ...................2...................

%C 4 3

%C 16......../ \........8 9......../ \........6

%C / \ / \ / \ / \

%C / \ / \ / \ / \

%C / \ / \ / \ / \

%C 256 32 64 12 81 18 36 5

%C 65536 512 1024 48 4096 128 144 24 6561 162 324 27 1296 72 25 10

%C etc.

%C This is the mirror image of the tree in A334860.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(0) = 1, and then after, a(2n) = a(n)^2, a(2n+1) = A334747(a(n)).

%F a(n) = A225546(A005940(1+n)).

%F For all n >= 0, A048675(a(n)) = A087808(n).

%o (PARI)

%o A334747(n) = { my(c=core(n), m=n); forprime(p=2, , if(c % p, m*=p; break, m/=p)); m; }; \\ From A334747

%o A334866(n) = if(!n,1,if(!(n%2),A334866(n/2)^2,A334747(A334866((n-1)/2))));

%Y Cf. A334865 (inverse permutation), A334860 (mirror image).

%Y Composition of permutations A005940 and A225546.

%Y Cf. A000290, A048675, A087808, A334747, A334870.

%Y Cf. A001146 (left edge of the tree), A019565 (right edge), A334110 (the left children of the right edge).

%K nonn,tabf

%O 0,2

%A _Antti Karttunen_, Jun 08 2020