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%I #17 Jun 05 2014 01:35:04
%S 1,4,12,2,32,8,84,6,19,24,220,3,18,50,63,53,564,13,9,138,49,128,162,
%T 10,31,136,38,365,1448,36,25,5,351,126,332,30,414,27,81,82,348,99,931,
%U 103,86,3699,96,929,21,14,64,223,16,79,892,210,325,847,80,265,1056,72,15,51,208,212,884,221,256
%N a(1)=1, and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))).
%C Self-inverse permutation of natural numbers.
%C Shares with A088609 the property that after 1, positions indexed by squarefree numbers larger than one, A005117(n+1): 2, 3, 5, 6, 7, 10, 11, 13, 14, ... contain only nonsquarefree numbers A013929: 4, 8, 9, 12, 16, 18, 20, 24, ..., and vice versa. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, thus implementing a kind of "deep" variant of A088609. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 is entangled with complementary pair A013929/A005117.
%H Antti Karttunen, <a href="/A243347/b243347.txt">Table of n, a(n) for n = 1..520</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+a(A057627(n))), otherwise, a(n) = A013929(a(A013928(n))). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929.]
%F For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2, where mu is Moebius mu (A008683). [Note: Permutation A088609 satisfies the same condition.]
%o (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o (definec (A243347 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (A005117 (+ 1 (A243347 (A057627 n))))) (else (A013929 (A243347 (A013928 n))))))
%Y Cf. A008966, A005117, A013929, A013928, A057627, A088609, A243348.
%Y Similar permutations: A236854, A235491, A243343-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.
%K nonn
%O 1,2
%A _Antti Karttunen_, Jun 03 2014
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