A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.

1, 1, 2, 2, 1, 3, 1, 3, 2, 2, 4, 3, 2, 2, 2, 4, 1, 4, 3, 1, 4, 2, 4, 5, 1, 2, 3, 1, 3, 7, 3, 3, 2, 6, 1, 3, 4, 3, 2, 4, 1, 7, 1, 3, 1, 8, 9, 2, 1, 3, 4, 1, 4, 4, 4, 4, 1, 3, 2, 2, 6, 8, 3, 1, 2, 10, 3, 5, 1, 1, 5, 4, 3, 3, 3, 3, 6, 3, 5, 7, 1, 6, 1, 5, 2, 4, 5

Offset: 1

Author

Gus Wiseman, May 29 2024

Keywords

• nonn

Examples

			This is the sequence of row-lengths of A005117 treated as a triangle with row-sums A373197:
   2
   3
   5   6
   7  10
  11
  13  14  15
  17
  19  21  22
  23  26
  29  30
  31  33  34  35
  37  38  39
  41  42
  43  46
  47  51
  53  55  57  58
		

Crossrefs

Counting all numbers (not just squarefree) gives A001223, sum A371201.

For composite instead of squarefree we have A046933.

For squarefree numbers (A005117) between primes:

- sum is A373197

- length is A373198 (this sequence) = A061398 - 1

- min is A000040

- max is A112925, opposite A112926

For squarefree numbers between powers of two:

- sum is A373123

- length is A077643, partial sums A143658

- min is A372683, delta A373125, indices A372540, firsts of A372475

- max is A372889, delta A373126

For primes between powers of two:

- sum is A293697 (except initial terms)

- length is A036378

- min is A104080 or A014210, indices A372684 (firsts of A035100)

- max is A014234, delta A013603

Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).

Cf. A049093, A049094, A076259, A077641.

Programs

Mathematica

Table[Length[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,100}]

Python

from math import isqrt
from sympy import prime, nextprime, mobius
def A373198(n):
    p = prime(n)
    q = nextprime(p)
    r = isqrt(p-1)+1
    return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r)) # Chai Wah Wu, Jun 01 2024

Formula

a(n) = A061398(n) + 1.