|
|
|
|
1, 2, 3, 4, 7, 5, 19, 6, 9, 11, 53, 10, 131, 23, 13, 8, 311, 15, 719, 22, 29, 59, 1619, 14, 49, 137, 21, 46, 3671, 17, 8161, 12, 61, 313, 37, 25, 17863, 727, 139, 26, 38873, 31, 84017, 118, 39, 1621, 180503, 20, 361, 77, 317, 274, 386093, 33, 71, 58, 733, 3673, 821641, 34, 1742537, 8167, 87, 18, 151, 67, 3681131, 626, 1627, 41, 7754077, 35, 16290047
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
A122111 and A225546 are both self-inverse permutations of the positive integers based on prime factorizations, and they share further common properties. For instance, they map the prime numbers to powers of 2: A122111 maps the k-th prime to 2^k, whereas A225546 maps it to 2^2^(k-1).
In composing these permutations, this sequence maps the squarefree numbers, as listed in A019565, to the prime numbers in increasing order; and the list of powers of 2 to the "normal" numbers (A055932), as listed in A057335.
|
|
LINKS
|
|
|
FORMULA
|
Alternative definition: (Start)
(End)
For n in A003159, a(2n) = b(a(n)), where b(1) = 2, b(n) = A253550(n), n >= 2.
More generally, a(A334747(n)) = b(a(n)).
|
|
EXAMPLE
|
In this set of examples we consider [a(n)] as a function a(.) with an inverse, a^-1(.).
First, a table showing mapping of the powers of 2:
n a^-1(2^n) = 2^n = a(2^n) =
0 (1) 1 1
1 2 2 2
2 4 4 4
3 16 8 6
4 256 16 8
5 65536 32 12
6 4294967296 64 18
...
Next, a table showing mapping of the squarefree numbers, as listed in A019565 (a lexicographic ordering by prime factors):
0 1 1 (1) (1)
1 2 2 2 2
2 3 3 3 3
3 8 6 5 7
4 6 5 7 19
5 12 10 11 53
6 18 15 13 131
7 128 30 17 311
8 5 7 19 719
9 24 14 23 1619
...
(End)
|
|
CROSSREFS
|
Lists of sets (S_1, S_2, ... S_j) related by the bijection defined by the sequence: (A000290\{0}, {1} U A070003), ({1} U A001146, A000079, A055932), ({1} U A335740, A005408, A066207), (A337533, A005117, A008578, {1} U A033844).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|