A375907 Lexicographically earliest sequence where a(n) is the length of the n-th block of distinct integers sharing a prime factor.

2, 4, 3, 6, 2, 4, 3, 6, 2, 3, 6, 2, 4, 8, 10, 3, 6, 5, 10, 2, 4, 3, 6, 2, 3, 6, 2, 4, 8, 10, 3, 6, 5, 10, 2, 3, 6, 2, 4, 8, 10, 3, 6, 5, 10, 2, 4, 3, 6, 2, 4, 8, 10, 5, 15, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 2, 4, 6, 5, 10, 2, 4, 6, 3, 2, 4, 6, 3, 9, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 2, 4, 3, 6, 2, 4

Offset: 1

Author

Bryle Morga, Sep 02 2024

Keywords

• nonn

Comments

Does every positive integer > 1 appear eventually?

Examples

			The sequence and blocks begin
  n    = 1 2  3 4 5 6  7 8 9 ...
  a(n) = 2,4, 3,6,2,4, 3,6,2, 3,6,2,4,8,10, 3,6, ...
  block  \-/  \-----/  \---/  \----------/  \-/
  length  2      4       3         6         2   ...
a(7) = 3 is the start of a new block since it has no common factor with the preceding a(6) = 4.
The block lengths are the sequence itself.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Programs

Python

from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    n, an, alst = 1, 2, [-3]
    while True:
        b = [next(i for i in count(2) if gcd(i, alst[-1])==1)]
        for i in range(an-1):
            b += [next(j for j in count(2) if j not in b and gcd(j, b[-1])!=1)]
        yield from b
        alst.extend(b)
        n, an = n+1, alst[n+1]
print(list(islice(agen(), 95))) # Michael S. Branicky, Sep 02 2024