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.

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

0,3

Comments

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!).

Links

Antti Karttunen, Table of n, a(n) for n = 0..10080

Index entries for sequences related to Matula-Goebel numbers

Index entries for sequences that are permutations of the natural numbers

Formula

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.

Prog

(Scheme, with Antti Karttunen's IntSeq-library)
(definec (A235201 n) (cond ((< n 3) n) ((= n 3) 4) ((zero? (modulo n 4)) (* 3 (A235201 (/ n 4)))) ((= 1 (A010051 n)) (A000040 (A235201 (A000720 n)))) (else (reduce * 1 (map A235201 (ifactor n))))))

Crossrefs

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.

Sequence in context: A021808 A365215 A225252 * A292958 A235493 A105081

Adjacent sequences: A235198 A235199 A235200 * A235202 A235203 A235204

Keyword

nonn,mult

Author

Antti Karttunen, Jan 11 2014

Status

approved