

A235489


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


9



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n, and is selfinverse. 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 "signaturepermutation" for a bijection of nonoriented rooted trees, mapped through MatulaGoebel 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 MatulaGoebelencoded by 9 (see above, or equally: replaced by 2k twoedgeslong branches encoded by 3), and one or two leftover single edges, if present, will stay as they are. Vice versa, any terminal configuration in the tree that consists of more than one twoedgeslong branches next to each other (in "parallel") will be transformed so that maximal even number (2k) of those doubleedges will be replaced by 3k single edges, and an extra odd doubleedge, if present, will stay as it is.
Note how in contrast to A235487, A235201 and A235199, this bijection is not sizepreserving (the number of edges will change), which has implications when composing this with other such permutations (cf. e.g. A235493/A235494).


LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to MatulaGoebel numbers
Index entries for sequences that are permutations of the natural numbers


FORMULA

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.


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary)
(definec (A235489 n) (cond ((< n 8) n) ((zero? (modulo n 8)) (* 9 (A235489 (/ n 8)))) ((zero? (modulo n 9)) (* 8 (A235489 (/ n 9)))) ((= 1 (A010051 n)) (A000040 (A235489 (A000720 n)))) (else (reduce * 1 (map A235489 (ifactor n))))))


CROSSREFS

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.
Sequence in context: A031978 A304481 A222254 * A065306 A065307 A207334
Adjacent sequences: A235486 A235487 A235488 * A235490 A235491 A235492


KEYWORD

nonn,mult


AUTHOR

Antti Karttunen, Jan 11 2014


STATUS

approved



