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A285655 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 distinct prime factors. 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Apr 23 2017
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)