A285848 Lexicographically earliest sequence of distinct positive terms such that for any n>0, i > j >=0, gcd(a^i(n), a^j(n)) = 1 (where a^k denotes the k-th iteration of a).

1, 3, 5, 7, 11, 13, 9, 15, 17, 19, 23, 25, 29, 27, 31, 21, 37, 35, 33, 39, 41, 43, 47, 49, 53, 45, 55, 51, 59, 61, 67, 57, 71, 63, 73, 65, 79, 69, 77, 81, 83, 85, 75, 87, 89, 91, 97, 95, 101, 93, 103, 99, 107, 109, 113, 111, 115, 105, 119, 121, 127, 117, 125

Offset

1,2

Comments

For any n>0, n and a(n) are coprime.

There is only one fixed point: a(1) = 1.

All terms are odd.

All terms > 1 have an even ancestor (if n > 1 then a(n) = a^i(2*j) for some i >= 0 and j > 0).

If n > 1, then a(n) > n.

This can be proved by induction, by considering u(n) = least odd term not seen among {a(1), ..., a(n-1)}, and noticing also that u(2*n) > 2*n.

The derived sequence b=(a+1)/2 is a permutation of the natural numbers.

The first terms of the orbit of 2 are: 2, 3, 5, 11, 23, 47, 97, 197, 401, 809, 1627, 3259, 61*107, 13063, 7*3733, 13*4021, 19*5503, 163*1283, 29*14423, 83*10079, 929*1801.

Conjecturally, a(n) ~ 2*n.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A285848

Example

a(1) = 1 is appropriate.
a(2) must be coprime to 2, and differ from 1; a(2) = 3 is appropriate.
a(3) must be coprime to 3 and 2, and differ from 1 and 3; a(3) = 5 is appropriate.
a(4) must be coprime to 4, and differ from 1, 3 and 5; a(4) = 7 is appropriate.

Crossrefs

Sequence in context: A282810 A102941 A114235 * A086527 A361445 A161554

Adjacent sequences: A285845 A285846 A285847 * A285849 A285850 A285851

Keyword

nonn

Author

Rémy Sigrist, Apr 27 2017

Status

approved