OFFSET
0,2
COMMENTS
This sequence can be represented as a binary tree. Each child to the left is obtained by applying A253550 to the parent, and each child to the right is obtained by applying A253560 to the parent:
1
|
...................2...................
3 4
5......../ \........9 6......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7 25 15 27 10 18 12 16
11 49 35 125 21 75 45 81 14 50 30 54 20 36 24 32
etc.
Sequence A253563 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) gives the distance of n from 1 in all these trees. Of these four trees, this is the one where the left child is always smaller than the right child.
Note that the indexing of sequence starts from 0, although its range starts from one.
The term a(n) is the Heinz number of the adjusted partial sums of the n-th composition in standard order, where (1) the k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again, (2) the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), and (3) we define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts. See formula for a simplification. A triangular form is A242628. The inverse is A253566. The non-adjusted version is A358170. - Gus Wiseman, Dec 17 2022
LINKS
FORMULA
As a composition of related permutations:
Other identities and observations. For all n >= 0:
a(2n+1) - a(2n) > 0. [See the comment above.]
If n = 2^(x_1)+...+2^(x_k) then a(n) = Product_{i=1..k} prime(x_k-x_{i-1}-k+i) where x_0 = 0. - Gus Wiseman, Dec 23 2022
EXAMPLE
From Gus Wiseman, Dec 23 2022: (Start)
This represents the following bijection between compositions and partitions. The n-th composition in standard order together with the reversed prime indices of a(n) are:
0: () -> ()
1: (1) -> (1)
2: (2) -> (2)
3: (1,1) -> (1,1)
4: (3) -> (3)
5: (2,1) -> (2,2)
6: (1,2) -> (2,1)
7: (1,1,1) -> (1,1,1)
8: (4) -> (4)
9: (3,1) -> (3,3)
10: (2,2) -> (3,2)
11: (2,1,1) -> (2,2,2)
12: (1,3) -> (3,1)
13: (1,2,1) -> (2,2,1)
14: (1,1,2) -> (2,1,1)
15: (1,1,1,1) -> (1,1,1,1)
(End)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]-1]+1, {n, 0, 60}] (* Gus Wiseman, Dec 17 2022 *)
PROG
CROSSREFS
KEYWORD
AUTHOR
Antti Karttunen, Jan 03 2015
STATUS
approved