A100113 a(n) = n if n <= 2, otherwise (smallest squarefree number m not occurring earlier such that gcd(m, a(n-1)) > 1).

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, 38, 19, 57, 66, 46, 23, 69, 78, 58, 29, 87, 93, 31, 62, 74, 37, 111, 102, 82, 41, 123, 105, 77, 91, 119, 85, 95, 110, 86, 43, 129, 114, 94, 47, 141, 138, 106, 53, 159, 165, 115, 130

Offset: 1

Author

Reinhard Zumkeller, Nov 07 2004

Keywords

• nonn

Comments

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);

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

Comments from N. J. A. Sloane, Oct 29 2020: (Start)

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.

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.

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.

(End)

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000 (terms 1..623 from N. J. A. Sloane)
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A005117, A064413, A100112, A100114.