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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?

Please see the comments at A234743. The observations given there apply also here.

Multiplicative because A234840 is.

Question: Are all terms positive? No negative terms in range 1 .. 2^17. Also (checked for n <= 2^17) the denominators seem to be given by A317932.

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

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.)

These are denominators for rational valued sequences that are obtained as "Dirichlet Square Roots" of sequences b that satisfy the condition b(3) = 2, and b(p) = odd number for any other primes p. For example, A064989, A065769 and A234840. - Antti Karttunen, Aug 31 2018

The original definition was: Denominators of the rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence. However, this definition depends on the conjecture given in A261179.

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=2, and is self-inverse. It swaps 3 & 4, maps any prime p_i with index i > 2 to p_{a(i)}, and lets the multiplicativity take care of the rest.

This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 3 and 4, wherever they occur as the terminal configurations anywhere in the tree:

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

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

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

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

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

.x.....<--->.....x..

.3...............4..

That is, the last two edges of any branch which ends with at least in two edges long unbranched stem, will be changed to a V-branch (two single edges in parallel). Vice versa, any terminal configuration in the tree that consists of more than one single edges next to each other (in "parallel") will be transformed so that maximal even number (2k) of those single edges will be combined to k unbranching stems of two edges, and an extra odd edge, if present, will stay as it is.

This permutation commutes with A235199, i.e. a(A235199(n)) = A235199(a(n)) for all n. This can be easily seen, when comparing the above bijection to the one described in A235199. Composition A235199 o A235201 works as a "difference" of these two bijections, swapping the above subconfigurations only when they do not occur alone at the tips of singular edges. (Which cases are encoded by Matula-Goebel numbers 5 and 7, the third and fourth prime respectively).

Permutation fixes n! for n=0, 1, 2, 4, 7.

Note that a(5!) = a(120) = 168 = 120+(2*4!) and a(8!) = a(40320) = 30240 = 40320-(2*7!).

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!).

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n, and is self-inverse. It swaps 8 & 9, maps any prime p_i with index i to p_{a(i)}, and lets the multiplicativity take care of the rest.

This can be viewed also as a "signature-permutation" for a bijection of non-oriented rooted trees, mapped through Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 8 and 9, wherever they occur as the terminal branches of the tree:

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

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

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

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

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

...x.....<--->.....x...

...8...............9...

Thus, any terminal configuration in the tree that consists of three or more single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be replaced by k subtrees Matula-Goebel-encoded by 9 (see above, or equally: replaced by 2k two-edges-long branches encoded by 3), and one or two left-over single edges, if present, will stay as they are. Vice versa, any terminal configuration in the tree that consists of more than one two-edges-long branches next to each other (in "parallel") will be transformed so that maximal even number (2k) of those double-edges will be replaced by 3k single edges, and an extra odd double-edge, if present, will stay as it is.

Note how in contrast to A235487, A235201 and A235199, this bijection is not size-preserving (the number of edges will change), which has implications when composing this with other such permutations (cf. e.g. A235493/A235494).

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=4, and is self-inverse. It swaps 7 & 8, maps any prime p_i with index i > 4 to p_{a(i)}, and lets the multiplicativity take care of the rest.

This can be viewed also as a "signature-permutation" for a bijection on non-oriented rooted trees, mapped through the Matula-Goebel numbers (cf. A061773). This bijection will swap the subtrees encoded by numbers 7 and 8, wherever they occur as the terminal configurations anywhere in the tree:

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

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

..\./..................

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

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

...x.....<--->.....x...

...7...............8...

Thus any branch of the tree that ends with three edges in Y-formation, will be transformed so that those three edges will emanate "in parallel" from the same vertex. Vice versa, any terminal configuration in the tree that consists of more than two single edges next to each other (in "parallel") will be transformed so that maximal 3k number of those single edges will be transformed to k Y-formations, and one or two left-over edges, if present, will stay as they are.