A375707 First differences minus 1 of nonsquarefree numbers.

3, 0, 2, 3, 1, 1, 3, 0, 1, 0, 3, 3, 3, 3, 0, 2, 0, 0, 1, 1, 1, 3, 2, 0, 3, 3, 2, 0, 3, 0, 2, 3, 1, 1, 3, 1, 0, 0, 3, 3, 3, 3, 0, 2, 0, 2, 0, 0, 1, 3, 2, 0, 3, 3, 2, 0, 1, 1, 0, 2, 3, 1, 1, 3, 0, 1, 0, 2, 0, 3, 3, 3, 0, 2, 3, 1, 1, 3, 2, 0, 3, 3, 3, 3, 0, 2, 3

Offset: 1

Author

Gus Wiseman, Sep 16 2024

Keywords

• nonn
• easy

Comments

Also the number of squarefree numbers between the nonsquarefree numbers A013929(n) and A013929(n+1).

Delete all 0's to get A120992.

The image is {0,1,2,3}.

Add 1 to all terms for A078147.

Examples

			The runs of squarefree numbers begin:
  (5,6,7)
  ()
  (10,11)
  (13,14,15)
  (17)
  (19)
  (21,22,23)
  ()
  (26)
  ()
  (29,30,31)
  (33,34,35)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of 0, 1, 2, 3 are A375709, A375710, A375711, A375712. This is a set partition of the positive integers into four blocks.

For runs of squarefree numbers:

- length: A120992, anti A373127

- min: A072284, anti A373408

- max: A373415, anti A007674

- sum: A373413, anti A373411

For runs of nonsquarefree numbers:

- length: A053797, anti A373409

- min: A053806, anti A373410

- max: A376164, anti A068781

- sum: A373414, anti A373412

A005117 lists the squarefree numbers, first differences A076259.

A013929 lists the nonsquarefree numbers, first differences A078147.

A046933 counts composite numbers between consecutive primes.

A073784 counts primes between consecutive composite numbers.

A093555 counts non-prime-powers between consecutive prime-powers.

Cf. A061398, A061399, A077641, A077643, A143658, A173143, A251092, A294242, A373125, A373126, A373573.

Programs

Mathematica

Differences[Select[Range[100],!SquareFreeQ[#]&]]-1

PARI

lista(nmax) = {my(prev = 4); for (n = 5, nmax, if(!issquarefree(n), print1(n - prev - 1, ", "); prev = n));} \\ Amiram Eldar, Sep 17 2024

Formula

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 6/(Pi^2-6) = 1.550546... . - Amiram Eldar, Sep 17 2024