A282841 Lexicographically earliest sequence of distinct terms such that a(n)=length of first run of multiples of prime(n) in this sequence.

1, 2, 3, 6, 4, 5, 10, 15, 7, 14, 21, 28, 35, 42, 8, 9, 11, 22, 33, 44, 12, 13, 26, 39, 52, 65, 16, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 18, 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 20, 23, 46, 69, 92, 115, 138, 161, 24, 25

Offset

1,2

Comments

For any i>0, let R_i denote the first run of multiples of prime(i) in this sequence; several such runs can overlap.

The following table gives the index of the first term belonging to k runs, alongside the corresponding runs, for small values of k:

k n Runs

-- -------- ----

0 1 None

1 2 R_1

2 1533 R_37 and R_38

3 7693674 R_1534 to R_1536

4 7706584 R_1534 to R_1537

5 7738564 R_1535 to R_1539

6 7751486 R_1535 to R_1540

7 37170235 R_3302 to R_3308

8 43960552 R_3551 to R_3558

9 44051293 R_3552 to R_3560

10 44145862 R_3553 to R_3562

11 44236717 R_3554 to R_3564

If i<j, then R_i starts before R_j.

For any prime p, there is a multiple of p in this sequence (see A282842).

Conjectures:

- For any k, there is a term belonging to k runs.

- This sequence is a permutation of the natural numbers.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..100000

Rémy Sigrist, Logarithmic scatterplot of the first hundred million terms

Rémy Sigrist, C# program for A282841

Example

a(1)=1 fits the definition.
a(2)=2 fits the definition and introduces R_1; R_1 has length a(1)=1: a(3) must not be a multiple of 2.
a(3)=3 fits the definition and the constraint on R_1.
a(3) introduces R_2; R_2 has length a(2)=2: a(4) must be a multiple of 3, whereas a(5) must not be a multiple of 3.
a(3)=6 fits the definition and the constraint on R_2.
a(4)=4 fits the definition and the constraint on R_2.
a(5)=5 fits the definition and introduces R_3.

Crossrefs

Cf. A282842, A282999.

Sequence in context: A361966 A358026 A331117 * A254106 A373323 A072637

Adjacent sequences: A282838 A282839 A282840 * A282842 A282843 A282844

Keyword

nonn

Author

Rémy Sigrist, Feb 22 2017

Status

approved