A288164 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+2) has at least 5 distinct prime factors.

1, 2, 2310, 1155, 3, 4, 770, 1365, 6, 8, 385, 1785, 12, 10, 455, 231, 18, 20, 595, 273, 22, 30, 105, 77, 26, 60, 165, 91, 14, 66, 195, 35, 28, 78, 255, 55, 38, 42, 210, 65, 11, 84, 390, 85, 7, 114, 330, 70, 13, 33, 420, 130, 17, 21, 462, 110, 5, 39, 546, 140

Offset

1,2

Comments

This sequence is a permutation of the natural numbers, with inverse A288799.

Conjecturally, a(n) ~ n.

For k >= 0, let f_k be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+k) has at least 5 distinct prime factors.

In particular, we have:

- f_0 = the numbers with at least 5 distinct prime factors,

- f_1 = A285487,

- f_2 = a (this sequence),

- f_3 = A288171.

If k > 0, then:

- f_k is a permutation of the natural numbers,

- f_k(i) = i for any i <= k,

- f_k(k+1) = A002110(5),

- conjecturally, f_k(n) ~ n.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Rémy Sigrist, C++ program for A288164

Index entries for sequences that are permutations of the natural numbers

Example

The first terms, alongside the primes p dividing a(n)*a(n+2), are:
n       a(n)    p
--      ----    --------------
1       1       2, 3, 5, 7, 11
2       2       2, 3, 5, 7, 11
3       2310    2, 3, 5, 7, 11
4       1155    2, 3, 5, 7, 11
5       3       2, 3, 5, 7, 11
6       4       2, 3, 5, 7,     13
7       770     2, 3, 5, 7, 11
8       1365    2, 3, 5, 7,     13
9       6       2, 3, 5, 7, 11
10      8       2, 3, 5, 7,         17
11      385     2, 3, 5, 7, 11
12      1785    2, 3, 5, 7,         17
13      12      2, 3, 5, 7,     13
14      10      2, 3, 5, 7, 11
15      455     2, 3, 5, 7,     13
16      231     2, 3, 5, 7, 11
17      18      2, 3, 5, 7,         17
18      20      2, 3, 5, 7,     13
19      595     2,    5, 7, 11,     17
20      273     2, 3, 5, 7,     13
21      22      2, 3, 5, 7, 11
22      30      2, 3, 5, 7, 11
23      105     2, 3, 5, 7,     13

Crossrefs

Cf. A002110, A285487, A288171, A288799 (inverse).

Sequence in context: A005800 A200170 A280899 * A133074 A280312 A276650

Adjacent sequences: A288161 A288162 A288163 * A288165 A288166 A288167

Keyword

nonn,look

Author

Rémy Sigrist, Jun 16 2017

Status

approved