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.

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

Author

Antti Karttunen, Jan 04 2014

Keywords

• nonn
• mult

Comments

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!).

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

Crossrefs

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.

Formula

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]