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

Author

Rémy Sigrist, Jun 16 2017

Keywords

• nonn
• look

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.

Examples

			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
		

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

Crossrefs

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