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).

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.

Links

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

Rémy Sigrist, Scatterplot of the first 10000000 terms (with prime terms highlighted)

Rémy Sigrist, C program for A320503

Index entries for sequences that are permutations of the natural numbers

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 A333841 A231236

Adjacent sequences: A320500 A320501 A320502 * A320504 A320505 A320506

Keyword

nonn

Author

Rémy Sigrist, Oct 13 2018

Status

approved