A375024 a(n) is the length of the largest sequence S of distinct integers in the range 1..n such that for any prime number p, any run of consecutive multiples of p in S has length exactly 2, and two consecutive terms in S have some common prime factor.

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 13, 13, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19

Offset

1,4

Comments

Sequences like A280864 can be split into segments of consecutive terms with properties similar to the sequences S that we are considering here.

Links

Table of n, a(n) for n=1..44.

Rémy Sigrist, C++ program

Formula

a(n) <= A373797(n).

a(p) = a(p-1) for any prime number p.

Example

Some solutions for small n:
  n   a(n)  Solution S
  --  ----  --------------------------------------------------------------
   1     1  1
   4     2  2,4
   6     3  2,6,3
  10     4  3,6,10,5
  15     7  3,6,10,15,12,14,7
  21    10  3,6,10,15,12,14,21,18,20,5
  33    13  3,6,10,15,12,14,21,18,22,33,24,20,5
  35    16  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,13
  39    19  3,6,10,15,12,14,21,18,20,35,28,22,33,24,26,39,36,34,17
  45    22  5,10,6,15,20,12,21,14,18,33,22,24,39,26,36,45,40,28,35,30,42,7

Prog

(C++) // See Links section.

Crossrefs

Cf. A280864, A283832, A373797.

Sequence in context: A146323 A071626 A182008 * A106457 A331852 A103128

Adjacent sequences: A375021 A375022 A375023 * A375025 A375026 A375027

Keyword

nonn,more

Author

Rémy Sigrist, Jul 28 2024

Status

approved