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A285296
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Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms is divisible by p^2 for some prime p.
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6
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1, 4, 2, 6, 3, 8, 5, 9, 7, 12, 10, 14, 16, 11, 18, 13, 20, 15, 21, 24, 17, 25, 19, 27, 22, 26, 28, 23, 32, 29, 36, 30, 33, 39, 40, 31, 44, 34, 38, 42, 35, 45, 37, 48, 41, 49, 43, 50, 46, 52, 47, 54, 51, 56, 53, 60, 55, 63, 57, 64, 58, 62, 66, 68, 59, 72, 61
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OFFSET
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1,2
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COMMENTS
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The sequence can always be extended with a number that is not squarefree (say a multiple of 4); after a term that is not squarefree, we can extend the sequence with the least unused number; as there are infinitely many multiples of 4, this sequence is a permutation of the natural numbers (with inverse A285297).
Conjecturally, a(n) ~ n.
This sequence has similarities with A075380: here we consider the product of consecutive terms, there the sum of consecutive terms.
For any k>0, let b_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms is divisible by p^k for some prime p; in particular we have:
- b_1 = A000027 (the natural numbers),
- b_2 = a (this sequence),
For any k>0, b_k is a permutation of the natural numbers.
For any k>0, b_k(1)=1 and b_k(2)=2^k.
Graphically, the sequences from b_2 to b_5 differ.
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LINKS
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EXAMPLE
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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
2 4 2
3 2 2
4 6 3
5 3 2
6 8 2
7 5 3
8 9 3
9 7 2
10 12 2
11 10 2
12 14 2
13 16 2
14 11 3
15 18 3
16 13 2
17 20 2, 5
18 15 3
19 21 2, 3
20 24 2
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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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