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

1,2

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

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Cf. A005117, A064413, A100112, A100114.

Sequence in context: A209664 A360451 A072647 * A300012 A302781 A303760

Adjacent sequences: A100110 A100111 A100112 * A100114 A100115 A100116

Keyword

nonn

Author

Reinhard Zumkeller, Nov 07 2004

Status

approved