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

1,2

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.

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

Example

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

Crossrefs

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

Sequence in context: A234735 A106771 A184502 * A074969 A066765 A067885

Adjacent sequences: A285652 A285653 A285654 * A285656 A285657 A285658

Keyword

nonn

Author

Rémy Sigrist, Apr 23 2017

Status

approved