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A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2*a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2*a(k).

%I A243343 #18 Jan 20 2020 21:42:56
%S A243343 1,3,7,2,15,5,31,6,14,11,63,4,13,29,23,30,127,10,9,62,27,59,47,12,28,
%T A243343 61,22,126,255,21,19,8,125,55,119,26,95,25,57,58,123,45,253,46,60,511,
%U A243343 43,254,20,18,39,124,17,54,251,118,111,239,53,94,191,51,24,56
%N A243343 a(1)=1; thereafter, if n is the k-th squarefree number (i.e., n = A005117(k)), a(n) = 1 + (2*a(k-1)); otherwise, when n is k-th nonsquarefree number (i.e., n = A013929(k)), a(n) = 2*a(k).
%C A243343 This is an instance of an "entanglement permutation", where two pairs of complementary subsets of natural numbers are interwoven with each other. In this case complementary pair A005117/A013929 (numbers which are squarefree/not squarefree) is entangled with complementary pair odd/even numbers (A005408/A005843).
%C A243343 Thus this shares with permutation A243352 the property that each term of A005117 is mapped bijectively to a unique odd number and likewise each term of A013929 is mapped (bijectively) to a unique even number. However, instead of placing terms into those positions in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself.
%C A243343 Are there any other fixed points than 1, 13, 54, 120, 1389, 3183, ... ?
%H A243343 Antti Karttunen, <a href="/A243343/b243343.txt">Table of n, a(n) for n = 1..10000</a>
%H A243343 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A243343 a(1) = 1; thereafter, if A008966(n) = 0 (i.e., n is a term of A013929, not squarefree), a(n) = 2*a(A057627(n)); otherwise a(n) = 2*a(A013928(n+1)-1)+1 (where A057627 and A013928(n+1) give the number of integers <= n divisible/not divisible by a square greater than one).
%F A243343 For all n, A000035(a(n)) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) modulo 2. The same property holds for A243352.
%o A243343 (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o A243343 (definec (A243343 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (* 2 (A243343 (A057627 n)))) (else (+ (* 2 (A243343 (- (A013928 (+ n 1)) 1))) 1))))
%Y A243343 Inverse: A243344.
%Y A243343 Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
%Y A243343 Similar permutations: A243352 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.
%K A243343 nonn
%O A243343 1,2
%A A243343 _Antti Karttunen_, Jun 03 2014