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(u*v) = a(u) * a(v).

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

Author

Antti Karttunen, Jan 11 2014

Keywords

• nonn
• mult

Comments

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.

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 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.

Formula

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).