A285487 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 5 distinct prime factors.

1, 2310, 2, 1155, 4, 1365, 6, 385, 12, 455, 18, 595, 22, 105, 26, 165, 14, 195, 28, 255, 38, 210, 11, 390, 7, 330, 13, 420, 17, 462, 5, 546, 10, 231, 20, 273, 30, 77, 60, 91, 66, 35, 78, 55, 42, 65, 84, 85, 114, 70, 33, 130, 21, 110, 39, 140, 51, 154, 15, 182

Offset: 1

Author

Rémy Sigrist, Apr 19 2017

Keywords

• nonn
• look

Comments

This sequence can always be extended with a multiple of 2310 = 2*3*5*7*11; after a term that has at least 5 distinct prime factors, we can extend the sequence with the least unused number; as there are infinitely many numbers with at least 5 distinct prime factors, this sequence is a permutation of the natural numbers (with inverse A285488).

Conjecturally, a(n) ~ n.

The first fixed points are: 1, 75, 295, 375, 1205, 1207, 1211, 6017, 19870, 19872, 19874, 19878, 19907, 19909.

Examples

			The first terms, alongside the primes p dividing a(n)*a(n+1), are:
n       a(n)    p
--      ----    --------------
1       1       2, 3, 5, 7, 11
2       2310    2, 3, 5, 7, 11
3       2       2, 3, 5, 7, 11
4       1155    2, 3, 5, 7, 11
5       4       2, 3, 5, 7,     13
6       1365    2, 3, 5, 7,     13
7       6       2, 3, 5, 7, 11
8       385     2, 3, 5, 7, 11
9       12      2, 3, 5, 7,     13
10      455     2, 3, 5, 7,     13
11      18      2, 3, 5, 7,        17
12      595     2,    5, 7, 11,    17
13      22      2, 3, 5, 7, 11
14      105     2, 3, 5, 7, 13
15      26      2, 3, 5,    11, 13
16      165     2, 3, 5, 7, 11
17      14      2, 3, 5, 7,     13
18      195     2, 3, 5, 7,     13
19      28      2, 3, 5, 7,        17
20      255     2, 3, 5,           17, 19
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..30000
• Rémy Sigrist, C++ program for A285487
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A285488.