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A320503
Lexicographically earliest sequence of distinct positive terms such that a(1) = 1, a(2) = 2, and for any n > 2, the least prime factor of a(n) does not exceed the prime next to the least prime factor of a(n-1).
3
1, 2, 3, 4, 6, 8, 9, 5, 7, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 25, 28, 30, 32, 33, 34, 36, 38, 39, 35, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 55, 49, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 58, 60, 62, 63, 64, 66, 68, 69, 65, 70
OFFSET
1,2
COMMENTS
More formally, for any n > 0, A055396(a(n+1)) <= A055396(a(n)) + 1.
This sequence is a variant of A320454.
This sequence is a permutation of the natural numbers, with inverse A320504; the reasoning to prove it is similar to that of A320454.
The prime numbers appear in ascending order as clusters in the sequence; the first prime clusters are:
- 2 terms: a(2) = 2, a(3) = 3,
- 2 terms: a(8) = 5, a(9) = 7,
- 12 terms: a(46) = 11, ..., a(57) = 53,
- 400 terms: a(2638) = 59, ..., a(3037) = 2861,
- 552398 terms: a(9149634) = 2879, ..., a(9702031) = 8207707.
The next prime cluster is expected to appear after the occurrence of the term 8207707^2.
The first known fixed points are: 1, 2, 3, 4, 10, 58, 84367, 2934331.
EXAMPLE
The first terms, alongside the least prime factor of a(n) and A055396(a(n)), are:
n a(n) lpf(a(n)) A055396(a(n))
-- ---- --------- -------------
1 1 N/A 0
2 2 2 1
3 3 3 2
4 4 2 1
5 6 2 1
6 8 2 1
7 9 3 2
8 5 5 3
9 7 7 4
10 10 2 1
11 12 2 1
12 14 2 1
13 15 3 2
14 16 2 1
15 18 2 1
MATHEMATICA
Nest[Append[#, Block[{k = 3, p}, While[Nand[Set[p, FactorInteger[k][[1, 1]]] <= NextPrime[#[[-1, -1]] ], FreeQ[#[[All, 1]], k ]], k++]; {k, p}]] &, {{1, 1}, {2, 2}}, 65][[All, 1]] (* Michael De Vlieger, Oct 17 2018 *)
PROG
(C) See Links section.
CROSSREFS
Cf. A055396, A320454, A320504 (inverse).
Sequence in context: A333220 A176798 A067118 * A138561 A376198 A333841
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Oct 13 2018
STATUS
approved