OFFSET
1,2
COMMENTS
The sequence can always be extended with a number that is not squarefree (say a multiple of 4); after a term that is not squarefree, we can extend the sequence with the least unused number; as there are infinitely many multiples of 4, this sequence is a permutation of the natural numbers (with inverse A285297).
Conjecturally, a(n) ~ n.
This sequence has similarities with A075380: here we consider the product of consecutive terms, there the sum of consecutive terms.
For any k>0, let b_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^k for some prime p; in particular we have:
- b_1 = A000027 (the natural numbers),
- b_2 = a (this sequence),
- b_3 = A285299,
- b_4 = A285386,
- b_5 = A285417.
For any k>0, b_k is a permutation of the natural numbers.
For any k>0, b_k(1)=1 and b_k(2)=2^k.
Graphically, the sequences from b_2 to b_5 differ.
LINKS
EXAMPLE
The first terms, alongside the primes p such that p^2 divides a(n)*a(n+1), are:
n a(n) p
-- ---- -
1 1 2
2 4 2
3 2 2
4 6 3
5 3 2
6 8 2
7 5 3
8 9 3
9 7 2
10 12 2
11 10 2
12 14 2
13 16 2
14 11 3
15 18 3
16 13 2
17 20 2, 5
18 15 3
19 21 2, 3
20 24 2
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Apr 16 2017
STATUS
approved