A335440 Lexicographically earliest sequence of distinct positive terms such that two distinct terms differ by at least 3 prime factors.

1, 8, 18, 50, 60, 64, 81, 98, 105, 144, 225, 242, 308, 338, 400, 429, 441, 480, 512, 546, 578, 625, 648, 722, 756, 784, 884, 935, 969, 1058, 1089, 1122, 1152, 1190, 1225, 1235, 1428, 1430, 1458, 1463, 1485, 1521, 1547, 1682, 1748, 1800, 1820, 1922, 1936, 2001

Offset: 1

Author

Rémy Sigrist, Jun 10 2020

Keywords

• nonn

Comments

In other words, for any distinct m and n, let a(m)/a(n) = u/v in reduced form, then bigomega(u) + bigomega(v) >= 3 (where bigomega corresponds to A001222(n), the number of distinct prime factors of n with multiplicity).

The variant where distinct terms differ by at least 1 prime factor simply corresponds to the positive numbers.

The variant where distinct terms differ by at least 2 prime factors corresponds to A028260.

No term is prime nor the square of a prime.

This sequence has similarities with A075926 and A333568; here we consider prime factors, there digits.

Examples

			The first terms, alongside their p-adic valuations for p = 2..11 (with dots instead of zeros), are:
  n   a(n)  v2  v3  v5  v7  v11
  --  ----  --  --  --  --  ---
   1     1   .   .   .   .    .
   2     8   3   .   .   .    .
   3    18   1   2   .   .    .
   4    50   1   .   2   .    .
   5    60   2   1   1   .    .
   6    64   6   .   .   .    .
   7    81   .   4   .   .    .
   8    98   1   .   .   2    .
   9   105   .   1   1   1    .
  10   144   4   2   .   .    .
  11   225   .   2   2   .    .
  12   242   1   .   .   .    2
  13   308   2   .   .   1    1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, C program for A335440

Crossrefs

Cf. A001222, A028260, A075926, A333568.

Programs

C

See Links section.