OFFSET
1,2
COMMENTS
Equivalently, for any n > 0, min(gpf(a(n)), gpf(a(n+1))) < max(lpf(a(n)), lpf(a(n+1))), where gpf = A006530 and lpf = A020639.
Also, for any n > 0, { a(n), a(n+1) } = { u, v } such that for any prime p and q, if p divides u and q divides v then p < q.
A p-smooth number is only divisible by prime numbers <= p, whereas a p-rough number is only divisible by prime numbers >= p.
This sequence is a permutation of the positive numbers, with inverse A296424:
- we can always extend the sequence with the least prime number that does not divide the product of the earlier terms,
- hence every prime number appear in the sequence, in increasing order,
- for any v > 0, v can appear after any prime number > v; as there are infinitely many such prime numbers, v will eventually appear.
The first known fixed points are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 108, 192, 1250.
Two consecutive terms are always coprime.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A296423
Rémy Sigrist, Scatterplot of the first 50000 terms
Rémy Sigrist, Colored scatterplot of the first 10000 terms
EXAMPLE
The first terms, alongside their distinct prime factors, are:
n a(n) distinct prime factors
-- ---- ----------------------
1 1 none
2 2 2
3 3 3
4 4 2
5 5 5
6 6 2, 3
7 7 7
8 8 2
9 9 3
10 11 11
11 10 2, 5
12 13 13
13 12 2, 3
14 17 17
15 14 2, 7
16 19 19
17 15 3, 5
18 16 2
19 21 3, 7
20 23 23
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Dec 12 2017
STATUS
approved