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A285655
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 distinct prime factors.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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_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.
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LINKS
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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