A330919 Lexicographically earliest sequence of distinct squarefree numbers such that for any n > 0, either a(n)/a(n+1) or a(n+1)/a(n) is a prime number.

1, 2, 6, 3, 15, 5, 10, 30, 210, 42, 14, 7, 21, 105, 35, 70, 770, 110, 22, 11, 33, 66, 330, 165, 55, 385, 77, 154, 462, 231, 1155, 2310, 30030, 2730, 390, 78, 26, 13, 39, 195, 65, 130, 910, 182, 91, 273, 546, 6006, 858, 286, 143, 429, 2145, 715, 1430, 4290

Offset: 1

Author

Rémy Sigrist, May 02 2020

Keywords

• nonn
• look

Comments

In other words, consecutive terms differ exactly by one prime factor.

This sequence has strong connections with A163252:

- here consecutive terms differ by one prime factor, there by one binary digit,

- for any n > 0, A163252(n-1) encodes in binary form the prime numbers appearing in a(n).

Odd indexed terms have an even number of prime factors and vice versa.

For any prime number p: as there are only finitely many squarefree numbers with greatest prime factor < p, eventually the sequence contains a multiple of p.

Examples

			The first terms, alongside their prime factors, are:
  n   a(n)  prime factors
  --  ----  -------------
   1     1
   2     2  2
   3     6  2, 3
   4     3     3
   5    15     3, 5
   6     5        5
   7    10  2,    5
   8    30  2, 3, 5
   9   210  2, 3, 5, 7
  10    42  2, 3,    7
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A330919

Crossrefs

Cf. A019565, A087207, A163252.

Programs

PARI

See Links section.

Formula

a(n) = A019565(A163252(n-1)).

A087207(a(n)) = A163252(n-1).