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

1,2

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.

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

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, 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
...

Crossrefs

Cf. A285296, A285576 (inverse).

Sequence in context: A371259 A037935 A159824 * A227168 A100252 A020340

Adjacent sequences: A285572 A285573 A285574 * A285576 A285577 A285578

Keyword

nonn,look

Author

Rémy Sigrist, Apr 22 2017

Status

approved