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.

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.

Links

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

Rémy Sigrist, PARI program for A296423

Rémy Sigrist, Scatterplot of the first 50000 terms

Rémy Sigrist, Colored scatterplot of the first 10000 terms

Index entries for sequences that are permutations of the natural numbers

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

Adjacent sequences: A296420 A296421 A296422 * A296424 A296425 A296426

Keyword

nonn

Author

Rémy Sigrist, Dec 12 2017

Status

approved