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A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.

%I A088610 #36 Mar 04 2024 01:35:00
%S A088610 1,4,2,8,3,9,5,12,6,16,7,18,10,20,11,24,13,25,14,27,15,28,17,32,19,36,
%T A088610 21,40,22,44,23,45,26,48,29,49,30,50,31,52,33,54,34,56,35,60,37,63,38,
%U A088610 64,39,68,41,72,42,75,43,76,46,80,47,81,51,84,53,88,55,90,57,92,58,96
%N A088610 Starting with n = 1, a(n) is the smallest squarefree number not included earlier if n is odd, else n is the smallest nonsquarefree number.
%C A088610 From _Antti Karttunen_, Jun 04 2014: (Start)
%C A088610 Squarefree (A005117) and nonsquarefree numbers (A013929) interleaved, the former at odd n and the latter at even n.
%C A088610 A243344 is a a "recursivized" variant of this permutation. Like this one, it also satisfies the given simple identity linking the parity of n with the Moebius mu-function. (End)
%H A088610 Antti Karttunen, <a href="/A088610/b088610.txt">Table of n, a(n) for n = 1..8192</a>
%H A088610 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A088610 From _Antti Karttunen_, Jun 04 2014: (Start)
%F A088610 a(2n) = A013929(n), a(2n-1) = A005117(n).
%F A088610 For all n, A008966(a(n)) = A000035(n), or equally, mu(a(n)) = n modulo 2, where mu is Moebius mu (A008683). (End)
%t A088610 With[{max = 100}, s = Select[Range[max], SquareFreeQ]; ns = Complement[Range[max], s]; Riffle[s[[1 ;; Length[ns]]], ns]] (* _Amiram Eldar_, Mar 04 2024 *)
%o A088610 (Scheme) (define (A088610 n) (if (even? n) (A013929 (/ n 2)) (A005117 (/ (+ 1 n) 2))))
%Y A088610 Inverse: A243352.
%Y A088610 Bisections: A005117, A013929.
%Y A088610 Similar permutations: A075378, A073846, A237056, A072061, A094077, A167151, A088609, A243344.
%Y A088610 Cf. A000035, A008966.
%K A088610 nonn
%O A088610 1,2
%A A088610 _Amarnath Murthy_, Oct 16 2003
%E A088610 More terms from _Ray Chandler_, Oct 18 2003