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A296423
Lexicographically earliest sequence of distinct positive terms such that, among any two consecutive terms, we have a p-smooth number and a (p+1)-rough number for some p > 0.
2
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 17, 14, 19, 15, 16, 21, 23, 18, 25, 24, 29, 20, 31, 22, 37, 26, 41, 27, 32, 33, 43, 28, 47, 30, 49, 36, 35, 48, 53, 34, 59, 38, 61, 39, 64, 45, 67, 40, 71, 42, 73, 44, 79, 46, 83, 50, 77, 54, 55, 72, 65, 81, 85, 89
OFFSET
1,2
COMMENTS
Equivalently, for any n > 0, min(gpf(a(n)), gpf(a(n+1))) < max(lpf(a(n)), lpf(a(n+1))), where gpf = A006530 and lpf = A020639.
Also, for any n > 0, { a(n), a(n+1) } = { u, v } such that for any prime p and q, if p divides u and q divides v then p < q.
A p-smooth number is only divisible by prime numbers <= p, whereas a p-rough number is only divisible by prime numbers >= p.
This sequence is a permutation of the positive numbers, with inverse A296424:
- we can always extend the sequence with the least prime number that does not divide the product of the earlier terms,
- hence every prime number appear in the sequence, in increasing order,
- for any v > 0, v can appear after any prime number > v; as there are infinitely many such prime numbers, v will eventually appear.
The first known fixed points are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 33, 108, 192, 1250.
Two consecutive terms are always coprime.
EXAMPLE
The first terms, alongside their distinct prime factors, are:
n a(n) distinct prime factors
-- ---- ----------------------
1 1 none
2 2 2
3 3 3
4 4 2
5 5 5
6 6 2, 3
7 7 7
8 8 2
9 9 3
10 11 11
11 10 2, 5
12 13 13
13 12 2, 3
14 17 17
15 14 2, 7
16 19 19
17 15 3, 5
18 16 2
19 21 3, 7
20 23 23
PROG
(PARI) See Links section.
CROSSREFS
Cf. A006530, A020639, A296424 (inverse).
Sequence in context: A113218 A296424 A065708 * A065650 A358386 A065649
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Dec 12 2017
STATUS
approved