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A320454
Lexicographically earliest sequence of distinct positive terms such that a(1) = 1, a(2) = 2, and for any n > 2, the greatest prime factor of a(n) does not exceed the prime next to the greatest prime factor of a(n-1).
3
1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 12, 15, 14, 11, 13, 16, 18, 20, 21, 22, 24, 25, 27, 30, 28, 32, 36, 40, 35, 33, 26, 17, 19, 23, 29, 31, 34, 38, 39, 42, 44, 45, 48, 50, 49, 54, 60, 56, 55, 52, 51, 57, 46, 58, 62, 37, 41, 43, 47, 53, 59, 61, 63, 64, 72, 75, 70
OFFSET
1,2
COMMENTS
More formally, for any n > 0, A061395(a(n+1)) <= A061395(a(n)) + 1.
This sequence is a permutation of the natural numbers, with inverse A320455:
- by induction: for any k > 0, every number with greatest prime factor prime(k) (where prime(k) denotes the k-th prime number) appear in the sequence:
- for k = 1: we can always choose a number with greatest prime factor 2, so eventually every number with greatest prime factor 2 will appear in the sequence,
- for any k > 1: provided every number with greatest prime factor prime(k) appear in the sequence: after a number with greatest prime factor prime(k), say w, we can always choose a number < w with greatest prime factor prime(k+1), so eventually every number with greatest prime factor prime(k+1) will appear in the sequence, QED.
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(6) = 5, a(7) = 7,
- 2 terms: a(14) = 11, a(15) = 13,
- 5 terms: a(32) = 17, ..., a(36) = 31,
- 7 terms: a(56) = 37, ..., a(62) = 61,
- 14 terms: a(139) = 67, ..., a(152) = 131,
- 26 terms: a(343) = 137, ..., a(368) = 271,
- 43 terms: a(745) = 277, ..., a(787) = 547,
- 85 terms: a(1893) = 557, ..., a(1977) = 1109,
- 145 terms: a(3963) = 1117, ..., a(4107) = 2221,
- 276 terms: a(10047) = 2237, ..., a(10322) = 4463,
- 506 terms: a(24973) = 4481, ..., a(25478) = 8951,
- 942 terms: a(44952) = 8963, ..., a(45893) = 17923.
See A320503 for a similar sequence.
EXAMPLE
The first terms, alongside the greatest prime factor of a(n) and A061395(a(n)), are:
n a(n) gpf(a(n)) A320454(a(n))
-- ---- --------- -------------
1 1 N/A 0
2 2 2 1
3 3 3 2
4 4 2 1
5 6 3 2
6 5 5 3
7 7 7 4
8 8 2 1
9 9 3 2
10 10 5 3
11 12 3 2
12 15 5 3
13 14 7 4
14 11 11 5
15 13 13 6
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
(PARI) See Links section.
CROSSREFS
Cf. A061395, A320455 (inverse), A320503.
Sequence in context: A130339 A350348 A058812 * A320455 A072796 A130374
KEYWORD
nonn,look
AUTHOR
Rémy Sigrist, Oct 13 2018
STATUS
approved