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 A243343 a(1)=1; thereafter, if n is 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). 15
 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, 61, 22, 126, 255, 21, 19, 8, 125, 55, 119, 26, 95, 25, 57, 58, 123, 45, 253, 46, 60, 511, 43, 254, 20, 18, 39, 124, 17, 54, 251, 118, 111, 239, 53, 94, 191, 51, 24, 56 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). Thus this shares with permutation A243352 the property that each term of A005117 is mapped bijectively to an unique odd number and likewise each term of A013929 is mapped (bijectively) to an 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. Are there any other fixed points than 1, 13, 54, 120, 1389, 3183, ... ? LINKS Antti Karttunen, Table of n, a(n) for n = 1..10000 FORMULA 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]. 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. PROG (Scheme, with memoizing definec-macro from Antti Karttunen's IntSeq-library) (definec (A243343 n) (cond ((<= n 1) n) ((zero? (A008966 n)) (* 2 (A243343 (A057627 n)))) (else (+ (* 2 (A243343 (- (A013928 (+ n 1)) 1))) 1)))) CROSSREFS Inverse: A243344. Cf. A005843, A005408, A008966, A005117, A013929, A013928, A057627. Similar permutations: A243352 (simple variant), A243345-A243346, A243347, A243287-A243288, A135141-A227413, A237126-A237427, A193231. Sequence in context: A163917 A266417 A260433 * A255565 A227351 A246377 Adjacent sequences:  A243340 A243341 A243342 * A243344 A243345 A243346 KEYWORD nonn AUTHOR Antti Karttunen, Jun 03 2014 STATUS approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)