A100113 - OEIS

%I #17 Oct 31 2020 18:59:39

%S 1,2,6,3,15,5,10,14,7,21,30,22,11,33,39,13,26,34,17,51,42,35,55,65,70,

%T 38,19,57,66,46,23,69,78,58,29,87,93,31,62,74,37,111,102,82,41,123,

%U 105,77,91,119,85,95,110,86,43,129,114,94,47,141,138,106,53,159,165,115,130

%N a(n) = n if n <= 2, otherwise (smallest squarefree number m not occurring earlier such that gcd(m, a(n-1)) > 1).

%C a(A100112(n)) and A100114(A100112(n)) define a pair of inverse permutations of the squarefree numbers: a(A100112(A100114(n))) = A100114(A100112(a(n))) = A005117(n);

%C A100115(n) = if n is squarefree then a(A100112(n)), otherwise n.

%C Comments from _N. J. A. Sloane_, Oct 29 2020 (Start)

%C An alternative definition is that this is the lexicographically earliest infinite sequence of distinct positive squarefree numbers with the property that gcd(a(n), a(n-1)) > 1 for n >= 3.

%C Described in this way, this is a squarefree version of the EKG sequence A064413, and it is easy to modify the proof that that sequence is a permutation of the positive integers so as to show that the present sequence is a permutation of the positive squarefree numbers, as claimed in the first comment.

%C Conjecture: With the three exceptions p = 2, 5, 13, and 31, when a prime p appears it is preceded by 2*p and followed by 3*p.

%C (End)

%H N. J. A. Sloane, <a href="/A100113/b100113.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%Y Cf. A005117, A064413, A100112, A100114.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Nov 07 2004