cp's OEIS Frontend

The permutation satisfies A008578(a(n)) = a(A008578(n)) for all n, and is self-inverse.

The sequence of fixed points begins as 0, 1, 6, 11, 29, 36, 66, 95, 107, 121, 149, 174, 216, 313, 319, 396, 427, ... and is itself multiplicative in a sense that if a and b are fixed points, then also a*b is a fixed point.

The records are 0, 1, 3, 9, 19, 61, 281, 361, 843, 1159, 1811, 3721, 5339, 5433, 17141, 78961, 110471, 236883, 325679, ...

and they occur at positions 0, 1, 2, 4, 5, 7, 13, 25, 26, 35, 37, 49, 65, 74, 91, 169, 259, 338, 455, ...

(Note how the permutations map squares to squares, and in general keep the prime signature the same.)

Composition with similarly constructed A235199 gives the permutations A234743 & A234744 with more open cycle-structure.

The result of applying a permutation of the prime numbers to the prime factors of n. - Peter Munn, Dec 15 2019

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=3 or 4, and is self-inverse. It swaps 5 & 7, maps all larger primes p_i (with index i > 4) to p_{a(i)}, and lets the multiplicativity take care of the rest.

It can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). The bijection will swap the subtrees encoded by primes 5 and 7, wherever they occur as the terminal branches of the tree:

....................

.o..................

.|..................

.o.............o...o

.|..............\./.

.o.....<--->.....o..

.|...............|..

.x...............x..

.5...............7..

That is, any branch which ends at least in three edges long unbranched stem, will be changed so that its last two edges will become V-branch. Vice versa, any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will be straightened to an unbranching stem of three edges.

This permutation commutes with A235201, i.e. a(A235201(n)) = A235201(a(n)) for all n.

Permutation fixes n! for n=0, 1, 2, 3, 4, 7, 8 and 9.

Note also that a(5!) = a(120) = 168 = 120+(2*4!) and a(10!) = 5080320 = 3628800+(4*9!).

Consider two self-inverse and multiplicative permutations, b and c defined as follows:

b(0)=0, b(1)=1, b(2)=3, b(3)=2, b(p_i) = p_{b(i+1)-1} for primes with index i > 2, and b(u*v) = b(u)*b(v) for u, v > 0.

c(n)=n if n < 4, c(5)=7 and c(7)=5, c(p_i) = p_{c(i)} for primes with index i > 4, and c(u*v) = c(u)*c(v) for u, v > 0.

This permutation is defined as their composition: a(n) = c(b(n)) = A235199(A234840(n)).

It is also multiplicative: a(u*v) = c(b(u*v)) = c(b(u)*b(v)) = c(b(u))*c(b(v)) = a(u)*a(v). For primes p_i with index i, a(p_i) = c(b(p_i)) = c(p_{b(i+1)-1}) = p_{c(b(i+1)-1)} = A000040(A235047(i)), except for cases i=8 and i=18, use 7 and 5, instead of 5 and 7.

Because 22 = 2*11, and 2 is in a two-cycle and 11 is in a three-cycle, 22 is in a cycle whose length is lcm(2,3) = 6: a(22)=51 (= a(2)*a(11) = 3*17), a(51)=202, a(202)=33, a(33)=34, a(34)=303, a(303)=22.

Among primes, there are at least fixed points (31), two-cycles (2 <-> 3), (37 <-> 1811), three-cycles: (11, 17, 101), (29, 43, 157), four-cycles: (5, 19, 7, 61), (41, 733, 359, 1091), eight-cycles: (47, 613, 2593, 1163, 1733, 409, 73, 131).

How long is the cycle beginning from 13, a(13)=433, a(433)=20693, a(20693)=? or from 23? (23, 173, 24043, ...)

Question: Are there any infinite cycles? If there are, what is the ratio of terms (primes) in finite cycles vs. infinite cycles?

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2 and n=4.

This permutation has only finite cycles: numbers 0, 1, 2, 3, ... are in the cycles of size: 1, 1, 1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 2, 4, 4, 4, 4, 5, 5, 1, 4, 4, 5, 4, 5, 4, 4, 2, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 7, 4, 4, 7, 4, 4, 4, 4, 4, 4, 7, 3, 7, 3, 1, ...

The first number with cycle size 1 (i.e., fixed point) is 0, the first in a 2-cycle is 3 (as a(3) = 4, a(4) = 3), the first in 3-cycle is 20, the first in 4-cycle is 5, the first in 5-cycle is 35, in 6-cycle 213, in 7-cycle 60, in 8-cycle and 9-cycle (no terms among 0..10080), the first in 10-cycle: 447, the first in 12-cycle: 220, in 14-cycle: 843, in 15-cycle: 2485, in 20-cycle: 385.

Please compare to the cycle structure of A235493/A235494.

Also of interest is the number of separate cycles (orbits) and fixed points among each A000081(n) rooted non-oriented trees when this bijection is applied to them, as trees encoded by Matula-Goebel numbers (cf. A061773).

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except for n=2.

In contrast to A235485/A235486 this permutation also contains infinite cycles.

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except for n=2.

In contrast to A235485/A235486 this permutation contains also infinite cycles.

The permutation satisfies A065091(a(n)) = a(A065091(n)) for all n >= 3, and is self-inverse: It swaps 3 & 5, maps all larger primes p_i (with index i > 3) to p_{a(i-1)+1}, and lets the multiplicativity take care of the rest.

Let b(n)=a(n), but with instead of a(3)=8 and a(4)=18, define b(3)=18 and b(4)=8 (i.e. otherwise same, but the values in positions 3 and 4 are swapped). The sequence b is then the permutation induced when A234744 is restricted to primes, and the indices of the reordered primes are collected: We have A049084(A234744(A000040(n))) = b(n) for all n. Or in other words, the permutation b completely determines the permutation A234744, because the latter is multiplicative. (Please see also comments there and at A234743.)