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

Author

Rémy Sigrist, Jun 19 2017

Keywords

• nonn
• look

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.

Examples

			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
		

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

Crossrefs

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