A061773 -id:A061773 - cp's OEIS frontend

A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 33, 65, 129, 257, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 41, 49, 50, 57, 66, 97, 130, 193, 258, 385, 513, 514, 769, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073

Offset: 1

Gus Wiseman, Nov 14 2022

We define the standard ordered rooted tree (SORT)-number of an unlabeled ordered rooted tree to be one plus the standard composition number (A066099) of the SORT-numbers of the branches, or 1 if there are no branches. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			Triangle begins:
   1
   2
   3   4
   5   6   7   8   9
  10  11  12  13  14  15  16  17  18  25  33  65 129 257
For example, the tree t = ((o,o),o) has branches (o,o) and o with SORT-numbers 4 and 1, and the standard composition number of (4,1) is 17, so t has SORT-number 18 and is found in row 5.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version for compositions is A000027.
Row-lengths are A000108.
The unordered version (using Matula-Goebel numbers) is A061773.
The version for Heinz numbers of partitions is A215366.
The row containing n is A358372(n).
A000081 counts unlabeled rooted trees, ranked by A358378.
A001263 counts unlabeled ordered rooted trees by nodes and leaves.
A358371 counts leaves in standard ordered rooted trees.
Cf. A004249, A005043, A061775, A109129, A284778, A358376, A358377.

stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
aotrank[t_]:=If[t=={},1,1+stcinv[aotrank/@t]];
aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Sort[aotrank/@aot[n]],{n,6}]

A005517 Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 625, 1125, 1875, 3125, 5625, 9375, 15625, 28125, 46875, 78125, 140625, 234375, 390625, 703125, 1171875, 1953125, 3515625, 5859375, 9765625, 17578125, 29296875, 48828125

Offset: 1

N. J. A. Sloane

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.

For n >= 3, this is also the minimum number of Hamiltonian paths in a strong tournament with n vertices (Busch). - Gordon Royle, Jan 24 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Arthur H. Busch, A Note on the Number of Hamiltonian Paths in Strong Tournaments, Electronic Journal of Combinatorics, N3 (2006).
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., 107, No. 19, 1994, 65-74.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to Matula-Goebel numbers
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (0,0,5).

Cf. A061773. See A005518 for the largest value of f(T).

a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif `mod`(n, 3) = 0 then 3*5^((1/3)*n-1) elif `mod`(n, 3) = 1 then 5^((1/3)*n-1/3) else 9*5^((1/3)*n-5/3) end if end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Apr 15 2012
A005517:=(-1-2*z-3*z**2+z**4)/(-1+5*z**3); # conjectured by Simon Plouffe in his 1992 dissertation

Join[{1,2},LinearRecurrence[{0,0,5},{3,5,9},40]] (* Harvey P. Dale, Feb 25 2012 *)
a[n_] := Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^((1/3)*n-1), Mod[n, 3] == 1, 5^((1/3)*n-1/3), True, 9*5^((1/3)*n-5/3)]; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Mar 06 2014, after Emeric Deutsch *)

a(1)=1; a(2)=2; a(n) = 3*5^((n-3)/3) if n=0 (mod 3); a(n)=5^((n-1)/3) if n>=4 and n=1 (mod 3); a(n)=9*5^((n-5)/3) if n>=5 and n = 2 (mod 3) (see the Gutman and Ivic 1994 paper). - Emeric Deutsch, Apr 15 2012
G.f.: z*(1+2*z+3*z^2-z^4)/(1-5*z^3) (conjectured by Simon Plouffe).
a(n+3) = 5*a(n) for n >= 3 under plausible assumptions about growth of prime numbers. - David W. Wilson, Jul 05 2001
A091233(n) = (A005518(n)-a(n))+1. - Antti Karttunen, May 24 2004

A005518 Largest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.

Original entry on oeis.org

1, 2, 4, 8, 19, 67, 331, 2221, 19577, 219613, 3042161, 50728129, 997525853, 22742734291, 592821132889, 17461204521323, 575411103069067, 21034688742654437, 846729487306354343

Offset: 1

N. J. A. Sloane

nonn

Let prime(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B, Vol. 29, No. 1 (1980), pp. 141-143.
I. Gutman and A. Ivić, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., Vol. 107, No. 19 (1994), pp. 65-74.
I. Gutman and A. Ivić, On Matula numbers, Discrete Math., Vol. 150, No. 1-3 (1996), pp. 131-142.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev., Vol. 10 (1968), p. 273.
Index entries for sequences related to Matula-Goebel numbers.
Index entries for sequences related to rooted trees.
Index entries for sequences related to trees.

Apart from initial terms, same as A057452.

Cf. A061773. See A005517 for the smallest value of f(T).

with(numtheory): a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif n = 3 then 4 elif n = 4 then 8 else ithprime(a(n-1)) end if end proc: seq(a(n), n = 1 .. 12); # Emeric Deutsch, Apr 15 2012

a[n_] := a[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, , Prime[a[n-1]]]; Table[a[n], {n, 1, 16}] (* _Jean-François Alcover, Mar 06 2014, after Emeric Deutsch *)

a(1)=1; a(2)=2; a(3)=4; a(4)=8; a(n) = the a(n-1)-th prime (see the Gutman and Ivic 1994 paper). - Emeric Deutsch, Apr 15 2012
Under plausible assumptions about the growth of the primes, for n >= 4, a(n+1) = a(n)-th prime and A005518(n) = A057452(n-3). - David W. Wilson, Jul 09 2001
A091233(n) = (a(n)-A005517(n))+1. - Antti Karttunen, May 24 2004

More terms from David W. Wilson, Jul 09 2001

a(17)-a(19) from Robert G. Wilson v, Mar 07 2017 using Kim Walisch's primecount

A209638 Sequence A209636 (or A209637) sorted into ascending order.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 24, 26, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 44, 48, 52, 53, 56, 58, 59, 62, 64, 67, 68, 71, 74, 76, 79, 80, 82, 86, 88, 89, 96, 101, 104, 106, 107, 109, 112, 116, 118, 124, 127, 128, 131, 134

Offset: 0

Antti Karttunen, Mar 11 2012

nonn

These are Matula-numbers (see A061773) for the rooted trees where no vertices with more than one non-leaf branch ever occur. In other words, natural numbers which are either some power of 2, or of the form 2^k * p_i, where k >= 0, and p_i is the i-th prime (A000040(i)), with i being one of the terms of this sequence.

Index entries for sequences related to Matula-Goebel numbers

Sorted version of A209636 and A209637.

Subset of A093641 and A122132.

from sympy import prime
def a(n):
    n = 2*n
    m = 1
    if n<2: return 1
    while n>1:
        if n%2==0:
            n//=2
            m*=2
        else:
            n=(n - 1)//2
            m=prime(m)
    return m
print(sorted([a(n) for n in range(101)])) # Indranil Ghosh, May 26 2017

A075166 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the prime factorization of n.

Original entry on oeis.org

0, 10, 1010, 1100, 101010, 101100, 10101010, 110100, 110010, 10101100, 1010101010, 10110100, 101010101010, 1010101100, 10110010, 111000, 10101010101010, 11001100, 1010101010101010, 1010110100, 1010110010

Offset: 1

Antti Karttunen, Sep 13 2002

Note that we recurse on the exponent + 1 for all other primes except the largest one in the factorization. Thus for 6 = 3^1 * 2^1 we construct a tree by joining trees 1 and 2 with a new root node, for 7 = 7^1 * 5^0 * 3^0 * 2^0 we join four 1-trees (single leaves) with a new root node, for 8 = 2^3 we add a single edge below tree 3 and for 9 = 3^2 * 2^0 we join trees 2 and 1, to get the mirror image of tree 6. Compare to Matula/Goebel numbering of (unoriented) rooted trees as explained in A061773.

			The rooted plane trees encoded here are:
.....................o...............o.........o...o..o.......
.....................|...............|..........\./...|.......
.......o....o...o....o....o.o.o..o...o.o.o.o.o...o....o...o...
.......|.....\./.....|.....\|/....\./...\|.|/....|.....\./....
*......*......*......*......*......*......*......*......*.....
1......2......3......4......5......6......7......8......9.....

A. Karttunen, Alternative Catalan Orderings (with the complete Scheme source)
A. Karttunen, Complete Scheme-program for computing this sequence.

Permutation of A063171. Same sequence shown in decimal: A075165. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A075167. Cf. A075171, A007088.

a(n) = A007088(A075165(n)) = A106456(A106442(n)). - Antti Karttunen, May 09 2005

A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 8, 5, 6, 16, 14, 17, 12, 19, 10, 28, 9, 11, 32, 13, 21, 20, 34, 53, 24, 49, 38, 64, 15, 43, 56, 59, 18, 68, 22, 35, 48, 37, 26, 76, 42, 67, 40, 29, 51, 112, 106, 107, 36, 25, 98, 44, 57, 23, 128, 119, 30, 52, 86, 31, 84, 131, 118, 80, 27, 133, 136, 41, 33

Offset: 0

Antti Karttunen, Jan 11 2014

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!).

Composition with A235487 gives A235485/A235486, composition with A235489 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A072026, A061773.

Multiplicative with a(3^k) = 2^(2k), a(2^(2k)) = 3^k, a(2^(2k+1)) = 2*3^k, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 14, 17, 12, 13, 10, 21, 16, 11, 18, 19, 28, 15, 34, 23, 24, 49, 26, 27, 20, 43, 42, 59, 32, 51, 22, 35, 36, 37, 38, 39, 56, 41, 30, 29, 68, 63, 46, 73, 48, 25, 98, 33, 52, 53, 54, 119, 40, 57, 86, 31, 84, 61, 118, 45, 64, 91, 102

Offset: 0

Antti Karttunen, Jan 04 2014

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!).

Composition with A234840 gives A234743 & A234744.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A235485/A235486, A235493/A235494, A091204/A091205, A072026, A061773.

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

A000035(a(n)) = A000035(n) = (n mod 2) for all n. [Even terms occur only on even indices and odd terms only on odd indices, respectively]

A235485 Permutation of natural numbers: a(n) = A235201(A235487(n)).

Original entry on oeis.org

0, 1, 2, 4, 3, 7, 8, 6, 5, 16, 14, 17, 12, 19, 9, 28, 10, 13, 32, 11, 21, 24, 34, 53, 20, 49, 38, 64, 18, 43, 56, 59, 15, 68, 26, 42, 48, 37, 22, 76, 35, 67, 36, 23, 51, 112, 106, 107, 40, 27, 98, 52, 57, 29, 128, 119, 30, 44, 86, 41, 84, 131, 118, 96, 25, 133, 136, 31, 39, 212, 63, 73, 80

Offset: 0

Antti Karttunen, Jan 11 2014

nonn

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).

Inverse: A235486. Cf. also A235201, A235487, A235493/A235494, A234743/A234744, A000081, A061773.

(define (A235485 n) (A235201 (A235487 n)))
;; Alternative implementation based on the given direct recurrence. Needs Antti Karttunen's IntSeq-library:
(definec (A235485 n) (cond ((< n 3) n) ((zero? (modulo n 3)) (* 4 (A235485 (/ n 3)))) ((zero? (modulo n 8)) (* 5 (A235485 (/ n 8)))) ((zero? (modulo n 4)) (* 3 (A235485 (/ n 4)))) ((zero? (modulo n 14)) (* 9 (A235485 (/ n 14)))) ((zero? (modulo n 49)) (* 27 (A235485 (/ n 49)))) ((zero? (modulo n 7)) (* 6 (A235485 (/ n 7)))) ((= 1 (A010051 n)) (A000040 (A235485 (A000720 n)))) (else (reduce * 1 (map A235485 (ifactor n))))))

a(n) = A235201(A235487(n)).
As a recurrence:
  a(0)=0, a(1)=1, a(2)=2,
  a(3*n) = 4*a(n),
  a(8*n) = 5*a(n),
  a(4*n) = 3*a(n) [when n is odd],
  a(14*n) = 9*a(n),
  a(49*n) = 27*a(n),
  a(7*n) = 6*a(n) [when n is odd and not divisible by 7],
  a(p_i) = p_{a(i)} for primes whose index i is neither 2 nor 4 [primes other than 3 or 7],
and
  a(u * v) = a(u) * a(v) for other composite cases.

Name and incorrect claim about multiplicativity corrected by Antti Karttunen, Feb 12 2018

A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 13, 14, 15, 18, 17, 16, 23, 20, 21, 22, 19, 27, 25, 26, 24, 28, 29, 30, 31, 36, 33, 34, 35, 32, 37, 46, 39, 45, 41, 42, 43, 44, 40, 38, 47, 54, 49, 50, 51, 52, 61, 48, 55, 63, 69, 58, 59, 60, 53, 62, 56, 81, 65, 66, 83, 68, 57, 70, 71, 72, 73, 74, 75, 92, 77, 78, 79, 90, 64

Offset: 0

Antti Karttunen, Jan 11 2014

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).

Composition with A235201 gives A235493/A235494.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235487 (swaps 7 & 8, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(3^(2k)) = 2^3k = 8^k, a(3^(2k+1)) = 3*2^3k, a(2^(3k)) = 3^2k = 9^k, a(2^(3k+1)) = 2*9^k, a(2^(3k+2)) = 4*9^k, a(p_i) = p_{a(i)} for primes with index i, and a(u*v) = a(u) * a(v) for composites other than 8 or 9.

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 16, 15, 14, 19, 18, 17, 20, 24, 22, 23, 21, 25, 26, 27, 32, 29, 30, 31, 28, 33, 38, 40, 36, 37, 34, 39, 35, 41, 48, 53, 44, 45, 46, 47, 42, 64, 50, 57, 52, 43, 54, 55, 56, 51, 58, 67, 60, 61, 62, 72, 49, 65, 66, 59, 76, 69, 80, 71, 63, 89, 74, 75, 68, 88, 78, 79, 70, 81

Offset: 0

Antti Karttunen, Jan 11 2014

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.

Composition with A235201 gives A235485/A235486.
List below gives similarly constructed permutations, which all force a swap of two small numbers, with (the rest of) primes permuted with the sequence itself and the new positions of composite numbers defined by the multiplicative property:
A234840 (swaps 2 & 3, conjugates A008578 back to itself).
A235200 (swaps 3 & 5, conjugates A065091 back to itself).
A235201 (swaps 3 & 4, conjugates A000040 back to itself).
A235199 (swaps 5 & 7, conjugates A000040 back to itself).
A235489 (swaps 8 & 9, conjugates A000040 back to itself).
Cf. also A000040, A010051, A000720, A091204/A091205, A061773.

Multiplicative with a(p_i) = p_{a(i)} for primes with index i <> 4, a(7) = 8, a(2^(3k)) = 7^k, a(2^(3k+1)) = 2*7^k, a(2^(3k+2)) = 4*7^k, and for other composites, a(u * v) = a(u) * a(v).

cp's OEIS Frontend

A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.

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A005517 Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.

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A005518 Largest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.

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A209638 Sequence A209636 (or A209637) sorted into ascending order.

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A075166 Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the prime factorization of n.

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A235201 Self-inverse and multiplicative permutation of integers: a(0)=0, a(1)=1, a(2)=2, a(3)=4 and a(4)=3, a(p_i) = p_{a(i)} for primes with index i > 2, and for composites > 4, a(u * v) = a(u) * a(v) for u, v > 0.

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A235199 Self-inverse and multiplicative permutation of integers: For n < 4, a(n)=n, a(5)=7 and a(7)=5, a(p_i) = p_{a(i)} for primes with index i > 4, and a(u * v) = a(u) * a(v) for u, v > 0.

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A235485 Permutation of natural numbers: a(n) = A235201(A235487(n)).

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A235489 Self-inverse and multiplicative permutation of integers: For n < 8, a(n) = n, a(8)=9 and a(9)=8, a(p_i) = p_{a(i)} for primes with index i, and for composites > 9, a(uv) = a(u) a(v).

A235487 Self-inverse and multiplicative permutation of integers: For n < 7, a(n)=n, a(7)=8 and a(8)=7, a(p_i) = p_{a(i)} for primes with index i <> 4, and for composites > 8, a(uv) = a(u) a(v).