A285575 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for at least two distinct primes p.

1, 36, 2, 18, 4, 9, 8, 25, 12, 3, 24, 6, 30, 10, 20, 5, 40, 15, 45, 16, 27, 28, 7, 56, 14, 42, 21, 48, 33, 44, 11, 72, 13, 52, 26, 50, 22, 54, 32, 49, 60, 35, 63, 64, 75, 39, 78, 66, 84, 51, 68, 17, 100, 19, 76, 38, 90, 34, 98, 46, 92, 23, 108, 29, 116, 58

Offset: 1

Author

Rémy Sigrist, Apr 22 2017

Keywords

• nonn
• look

Comments

The sequence can always be extended with a multiple of 36; after a multiple of 36, we can extend the sequence with the least unused number; as there are infinitely many multiples of 36, this sequence is a permutation of the natural numbers (with inverse A285576).

For any k>=0, let c_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^2 for at least k distinct primes p; in particular we have:

- c_0 = A000027 (the natural numbers),

- c_1 = A285296,

- c_2 = a (this sequence).

For any k>=0, c_k is a permutation of the natural numbers.

Examples

			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, 3
2       36      2, 3
3       2       2, 3
4       18      2, 3
5       4       2, 3
6       9       2, 3
7       8       2,    5
8       25      2,    5
9       12      2, 3
10      3       2, 3
11      24      2, 3
12      6       2, 3
13      30      2,    5
14      10      2,    5
15      20      2,    5
16      5       2,    5
17      40      2,    5
18      15         3, 5
19      45      2, 3
20      16      2, 3
...
115     160     2,    5
116     115     2, 3, 5
117     180     2, 3
...
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A285575
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A285296, A285576 (inverse).