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A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.

%I A088609 #17 Jun 08 2014 08:57:03
%S A088609 1,4,8,2,9,12,16,3,5,18,20,6,24,25,27,7,28,10,32,11,36,40,44,13,14,45,
%T A088609 15,17,48,49,50,19,52,54,56,21,60,63,64,22,68,72,75,23,26,76,80,29,30,
%U A088609 31,81,33,84,34,88,35,90,92,96,37,98,99,38,39,100,104,108,41,112,116
%N A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.
%C A088609 From _Antti Karttunen_, Jun 04 2014: (Start)
%C A088609 This is a self-inverse permutation (involution) of natural numbers.
%C A088609 After 1, nonsquarefree numbers occur (in monotonic order) at the positions given by squarefree numbers, A005117, and squarefree numbers occur (in monotonic order) at the positions given by their complement, nonsquarefree numbers, A013929.
%C A088609 (End)
%H A088609 Antti Karttunen, <a href="/A088609/b088609.txt">Table of n, a(n) for n = 1..10001</a>
%H A088609 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A088609 From _Antti Karttunen_, Jun 04 2014: (Start)
%F A088609 a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+A057627(n)), otherwise, a(n) =  A013929(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 A088609 For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2. [A property shared with A243347].
%F A088609 (End)
%o A088609 (define (A088609 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (A005117 (+ 1 (A057627 n)))) (else (A013929 (A013928 n))))) ;; _Antti Karttunen_, Jun 04 2014
%Y A088609 Cf. A026239, A088610-A243352, A243347.
%K A088609 nonn
%O A088609 1,2
%A A088609 _Amarnath Murthy_, Oct 16 2003
%E A088609 More terms from _Ray Chandler_, Oct 18 2003