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A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.

%I A243345 #17 Jan 23 2015 22:52:10
%S A243345 1,2,4,3,8,6,16,5,9,12,32,7,10,18,24,17,64,13,14,33,20,36,48,11,19,34,
%T A243345 25,65,128,26,28,15,66,40,72,21,96,22,38,37,68,50,130,49,35,256,52,
%U A243345 129,27,29,56,67,30,41,132,73,80,144,42,97,192,44,23,39,76,74,136,69,100
%N A243345 a(1)=1; thereafter, if n is k-th squarefree number [i.e., n = A005117(k)], a(n) = 2*a(k-1); otherwise, when n is k-th nonsquarefree number [i.e., n = A013929(k)], a(n) = 2*a(k)+1.
%C A243345 Any other fixed points than 1, 2, 6, 9, 135, 147, 914, ... ?
%C A243345 Any other points than 4, 21, 39, 839, 4893, 12884, ... where a(n) = n-1 ?
%H A243345 Antti Karttunen, <a href="/A243345/b243345.txt">Table of n, a(n) for n = 1..10000</a>
%H A243345 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A243345 a(1) = 1, and for n>1, if mu(n) = 0, a(n) = 1 + 2*a(A057627(n)), otherwise a(n) = 2*a(A013928(n)), where mu is Moebius mu function (A008683).
%F A243345 For all n > 1, A000035(a(n)+1) = A008966(n) = A008683(n)^2, or equally, a(n) = mu(n) + 1 modulo 2.
%o A243345 (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o A243345 (definec (A243345 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (+ 1 (* 2 (A243345 (A057627 n))))) (else (* 2 (A243345 (A013928 n))))))
%Y A243345 Inverse: A243346.
%Y A243345 Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627.
%Y A243345 Similar permutations: A243343-A243344, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.
%K A243345 nonn,look
%O A243345 1,2
%A A243345 _Antti Karttunen_, Jun 03 2014