A288923 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 prime factors (counted with multiplicity).

1, 64, 2, 32, 3, 48, 4, 16, 6, 24, 8, 12, 18, 20, 27, 28, 30, 36, 9, 40, 10, 54, 14, 56, 15, 60, 21, 72, 5, 80, 7, 96, 11, 108, 13, 112, 17, 120, 19, 128, 22, 81, 25, 84, 26, 88, 33, 90, 34, 100, 35, 104, 38, 126, 39, 132, 42, 44, 45, 50, 52, 63, 66, 68, 70

Offset

1,2

Comments

The number of prime factors counted with multiplicity is given by A001222.

This sequence is a permutation of the natural numbers, with inverse A288924.

Conjecturally, a(n) ~ n.

For a function g over the natural numbers and a constant K, let f(g,K) be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, g( f(g,K)(n) * f(g,K)(n+1) ) >= K. In particular we have:

- f(bigomega, 6) = a (this sequence), where bigomega = A001222,

- f(tau, 34) = A288921, where tau = A000005,

- f(omega, 5) = A285487, where omega = A001221,

- f(omega, 6) = A285655, where omega = A001221.

Some of these sequences have similar graphical features.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..20000

Rémy Sigrist, PARI program for A288923

Rémy Sigrist, Colored scatterplot of a(n) for n = 1..20000 (where the color is function of A001222(a(n)))

Index entries for sequences that are permutations of the natural numbers

Example

The first terms, alongside a(n) * a(n+1) and its number of prime divisors counted with multiplicity, are:
   n   a(n)   a(n)*a(n+1)   Bigomega
  --   ----   -----------   --------
   1     1         64           6
   2    64        128           7
   3     2         64           6
   4    32         96           6
   5     3        144           6
   6    48        192           7
   7     4         64           6
   8    16         96           6
   9     6        144           6
  10    24        192           7
  11     8         96           6
  12    12        216           6
  13    18        360           6
  14    20        540           6
  15    27        756           6
  16    28        840           6
  17    30       1080           7
  18    36        324           6
  19     9        360           6
  20    40        400           6

Crossrefs

Cf. A001221, A001222, A285487, A285655, A288921, A288924 (inverse).

Sequence in context: A302155 A351246 A371277 * A123964 A298923 A210114

Adjacent sequences: A288920 A288921 A288922 * A288924 A288925 A288926

Keyword

nonn,look

Author

Rémy Sigrist, Jun 19 2017

Status

approved