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

1,1

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.

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

Example

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

Crossrefs

Cf. A285487.

Sequence in context: A280659 A060231 A285487 * A361039 A051270 A046387

Adjacent sequences: A285741 A285742 A285743 * A285745 A285746 A285747

Keyword

nonn,look

Author

Rémy Sigrist, Apr 25 2017

Status

approved