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%I #28 Mar 07 2017 06:20:11
%S 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,
%T 26,27,20,43,42,59,32,51,22,35,36,37,38,39,56,41,30,29,68,63,46,73,48,
%U 25,98,33,52,53,54,119,40,57,86,31,84,61,118,45,64,91,102
%N 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.
%C 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.
%C 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:
%C ....................
%C .o..................
%C .|..................
%C .o.............o...o
%C .|..............\./.
%C .o.....<--->.....o..
%C .|...............|..
%C .x...............x..
%C .5...............7..
%C 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.
%C This permutation commutes with A235201, i.e. a(A235201(n)) = A235201(a(n)) for all n.
%C Permutation fixes n! for n=0, 1, 2, 3, 4, 7, 8 and 9.
%C Note also that a(5!) = a(120) = 168 = 120+(2*4!) and a(10!) = 5080320 = 3628800+(4*9!).
%H Antti Karttunen, <a href="/A235199/b235199.txt">Table of n, a(n) for n = 0..10080</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F 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.
%F 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]
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (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))))))
%Y Composition with A234840 gives A234743 & A234744.
%Y 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:
%Y A234840 (swaps 2 & 3, conjugates A008578 back to itself).
%Y A235200 (swaps 3 & 5, conjugates A065091 back to itself).
%Y A235201 (swaps 3 & 4, conjugates A000040 back to itself).
%Y A235487 (swaps 7 & 8, conjugates A000040 back to itself).
%Y A235489 (swaps 8 & 9, conjugates A000040 back to itself).
%Y Cf. also A000040, A010051, A000720, A235485/A235486, A235493/A235494, A091204/A091205, A072026, A061773.
%K nonn,mult
%O 0,3
%A _Antti Karttunen_, Jan 04 2014
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