A285655 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 distinct prime factors.

1, 30030, 2, 15015, 4, 19635, 6, 5005, 12, 6545, 18, 7315, 24, 7735, 22, 1365, 34, 1155, 26, 1785, 38, 2145, 14, 2805, 28, 3135, 42, 715, 84, 935, 78, 385, 102, 455, 66, 595, 114, 770, 39, 1190, 33, 910, 51, 1330, 69, 1430, 21, 1870, 57, 1540, 87, 1610, 93

Offset: 1

Author

Rémy Sigrist, Apr 23 2017

Keywords

• nonn

Comments

This sequence can always be extended with a multiple of 30030 = 2*3*5*7*11*13; after a term that has at least 6 distinct prime factors, we can extend the sequence with the least unused number; as there are infinitely many numbers with at least 6 distinct prime factors, this sequence is a permutation of the natural numbers (with inverse A285656).

Conjecturally, a(n) ~ n.

The first fixed points are: 1, 39, 1344, 1350, 3556, 3560, 5738, 6974, 15668585, 15668673, 15668787.

For any k>0, let d_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms has at least k distinct prime factors; in particular we have:

- d_1 = A000027 (the natural numbers),

- d_5 = A285487,

- d_6 = a (this sequence).

For any k>0:

- d_k is a permutation of the natural numbers,

- d_k(1) = 1 and d_k(2) = A002110(k),

- conjecturally: d_k(n) ~ n.

Examples

			The first terms, alongside the primes p dividing a(n)*a(n+1), are:
n       a(n)    p
--      ----    ------------------
1       1       2, 3, 5, 7, 11, 13
2       30030   2, 3, 5, 7, 11, 13
3       2       2, 3, 5, 7, 11, 13
4       15015   2, 3, 5, 7, 11, 13
5       4       2, 3, 5, 7, 11,     17
6       19635   2, 3, 5, 7, 11,     17
7       6       2, 3, 5, 7, 11, 13
8       5005    2, 3, 5, 7, 11, 13
9       12      2, 3, 5, 7, 11,     17
10      6545    2, 3, 5, 7, 11,     17
11      18      2, 3, 5, 7, 11,         19
12      7315    2, 3, 5, 7, 11,         19
13      24      2, 3, 5, 7,     13, 17
14      7735    2,    5, 7, 11, 13, 17
15      22      2, 3, 5, 7, 11, 13
16      1365    2, 3, 5, 7,     13, 17
17      34      2, 3, 5, 7, 11,     17
18      1155    2, 3, 5, 7, 11, 13
19      26      2, 3, 5, 7,     13, 17
20      1785    2, 3, 5, 7,         17, 19
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Scatterplot of the first 20 000 000 terms
• Rémy Sigrist, Scatterplot of the first difference of the first 20 000 000 terms
• Rémy Sigrist, C++ program for A285655
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000027, A002110, A285487, A285656 (inverse).