A285744 Lexicographically earliest sequence of distinct positive terms such that, for any n>0, n*a(n) has at least 5 distinct prime factors.

2310, 1155, 770, 1365, 462, 385, 330, 1785, 910, 231, 210, 455, 420, 165, 154, 1995, 390, 595, 510, 273, 110, 105, 546, 665, 714, 255, 1190, 195, 570, 77, 630, 2145, 70, 285, 66, 715, 660, 315, 140, 357, 690, 55, 780, 345, 182, 399, 798, 805, 858, 429, 130

Offset: 1

Author

Rémy Sigrist, Apr 25 2017

Keywords

• nonn
• look

Comments

If n has at least 5 distinct prime factors, then a(n) is 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.

The inverse of this sequence is the sequence itself.

The first fixed points are: 40755, 42966, 54285, 54740, 55965, 56070, 66045, 66066, 70035, 70350, 73815, 73920 (note that the fixed points have at least 5 distinct prime factors).

Conjecturally, a(n) ~ n.

This sequence has similarities with A285487: here n*a(n) has at least 5 distinct prime factors, there a(n)*a(n+1) has at least 5 distinct prime factors.

Examples

			The first terms, alongside the primes p dividing n*a(n), are:
n       a(n)    p
--      ----    --------------
1       2310    2, 3, 5, 7, 11
2       1155    2, 3, 5, 7, 11
3       770     2, 3, 5, 7, 11
4       1365    2, 3, 5, 7,     13
5       462     2, 3, 5, 7, 11
6       385     2, 3, 5, 7, 11
7       330     2, 3, 5, 7, 11
8       1785    2, 3, 5, 7,         17
9       910     2, 3, 5, 7,     13
10      231     2, 3, 5, 7, 11
11      210     2, 3, 5, 7, 11
12      455     2, 3, 5, 7,     13
13      420     2, 3, 5, 7,     13
14      165     2, 3, 5, 7, 11
15      154     2, 3, 5, 7, 11
16      1995    2, 3, 5, 7,             19
17      390     2, 3, 5,        13, 17
18      595     2, 3, 5, 7,         17
19      510     2, 3, 5,            17, 19
20      273     2, 3, 5, 7,     13
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..25000
• Rémy Sigrist, PARI program for A285744
• Rémy Sigrist, Colored scatterplot of the first 25000 terms
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A285487.