cp's OEIS Frontend

A243347 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))).

%I A243347 #17 Jun 05 2014 01:35:04
%S A243347 1,4,12,2,32,8,84,6,19,24,220,3,18,50,63,53,564,13,9,138,49,128,162,
%T A243347 10,31,136,38,365,1448,36,25,5,351,126,332,30,414,27,81,82,348,99,931,
%U A243347 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 A243347 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 A243347 Self-inverse permutation of natural numbers.
%C A243347 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 A243347 Antti Karttunen, <a href="/A243347/b243347.txt">Table of n, a(n) for n = 1..520</a>
%H A243347 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A243347 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 A243347 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 A243347 (Scheme, with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)
%o A243347 (definec (A243347 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (A005117 (+ 1 (A243347 (A057627 n))))) (else (A013929 (A243347 (A013928 n))))))
%Y A243347 Cf. A008966, A005117, A013929, A013928, A057627, A088609, A243348.
%Y A243347 Similar permutations: A236854, A235491, A243343-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231.
%K A243347 nonn
%O A243347 1,2
%A A243347 _Antti Karttunen_, Jun 03 2014