A334860 - OEIS

%I #24 Jun 11 2020 23:43:40

%S 1,2,3,4,6,9,8,16,5,36,18,81,12,64,32,256,10,25,72,1296,27,324,162,

%T 6561,24,144,128,4096,48,1024,512,65536,15,100,50,625,108,5184,2592,

%U 1679616,54,729,648,104976,243,26244,13122,43046721,20,576,288,20736,192,16384,8192,16777216,96,2304,2048,1048576,768,262144,131072,4294967296,30

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

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

%C                                       1

%C                                       |

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

%C                   3                                       4

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

%C        / \                 / \                 / \                 / \

%C       /   \               /   \               /   \               /   \

%C      /     \             /     \             /     \             /     \

%C     5       36         18       81         12       64         32      256

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

%C etc.

%C This is the mirror image of the tree in A334866.

%C Fermi-Dirac primes, A050376, occur at rightward growing branches that originate from primes situated at the left edge.

%C The tree illustrated in A163511 is expanded as x -> 2*x for the left child and x -> A003961(x) for the right child, while this tree is expanded as x -> A225546(2*A225546(x)) for the left child, and x -> A225546(A003961(A225546(x))) for the right child.

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

%F a(0) = 1, a(1) = 2; and for n > 0, a(2n) = A334747(a(n)), a(2n+1) = a(n)^2.

%F a(n) = A225546(A163511(n)).

%F For n >= 0, a(2^n) = A019565(1+n), a(2^((2^n)-1)) = A000040(1+n).

%F A334109(a(n)) = A334204(n).

%F It seems that for n >= 1, A048675(a(n)) = A135529(n) = A048675(A163511(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 A334860(n) = if(n<=1,1+n,if(!(n%2),A334747(A334860(n/2)),A334860((n-1)/2)^2));

%Y Cf. A000290, A225546, A334204, A334747, A334859 (inverse), A334866 (mirror image).

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

%Y Composition of permutations A163511 and A225546.

%K nonn,tabf

%O 0,2

%A _Antti Karttunen_, Jun 08 2020