A285417 -id:A285417 - cp's OEIS frontend

A285296 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.

1, 4, 2, 6, 3, 8, 5, 9, 7, 12, 10, 14, 16, 11, 18, 13, 20, 15, 21, 24, 17, 25, 19, 27, 22, 26, 28, 23, 32, 29, 36, 30, 33, 39, 40, 31, 44, 34, 38, 42, 35, 45, 37, 48, 41, 49, 43, 50, 46, 52, 47, 54, 51, 56, 53, 60, 55, 63, 57, 64, 58, 62, 66, 68, 59, 72, 61

Offset: 1

Rémy Sigrist, Apr 16 2017

nonn

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.

			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

Rémy Sigrist, Table of n, a(n) for n = 1..2000
Rémy Sigrist, PARI program for A285296
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A075380, A285297 (inverse).

cp's OEIS Frontend

A285296 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.

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