

A235199


Selfinverse 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.


11



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
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OFFSET

0,3


COMMENTS

The permutation satisfies A000040(a(n)) = a(A000040(n)) for all positive n except n=3 or 4, and is selfinverse. 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 "signaturepermutation" for a bijection of nonoriented rooted trees, mapped through MatulaGoebel 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 Vbranch. Vice versa, any branch of the tree that ends with three edges in Yformation, 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 MatulaGoebel numbers
Index entries for sequences that are permutations of the natural numbers


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]


PROG

(Scheme, with Antti Karttunen's IntSeqlibrary)
(definec (A235199 n) (cond ((< n 4) n) ((= n 5) 7) ((= n 7) 5) ((= 1 (A010051 n)) (A000040 (A235199 (A000720 n)))) (else (reduce * 1 (map A235199 (ifactor n))))))


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.
Sequence in context: A072028 A269377 A072026 * A270426 A270425 A085161
Adjacent sequences: A235196 A235197 A235198 * A235200 A235201 A235202


KEYWORD

nonn,mult


AUTHOR

Antti Karttunen, Jan 04 2014


STATUS

approved



