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).

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Scatterplot of the first 50000 terms (with prime terms highlighted)

Rémy Sigrist, PARI program for A320454

Index entries for sequences that are permutations of the natural numbers

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

Adjacent sequences: A320451 A320452 A320453 * A320455 A320456 A320457

Keyword

nonn,look

Author

Rémy Sigrist, Oct 13 2018

Status

approved