A061398 Number of squarefree integers between prime(n) and prime(n+1).

0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 3, 2, 1, 1, 1, 3, 0, 3, 2, 0, 3, 1, 3, 4, 0, 1, 2, 0, 2, 6, 2, 2, 1, 5, 0, 2, 3, 2, 1, 3, 0, 6, 0, 2, 0, 7, 8, 1, 0, 2, 3, 0, 3, 3, 3, 3, 0, 2, 1, 1, 5, 7, 2, 0, 1, 9, 2, 4, 0, 0, 4, 3, 2, 2, 2, 2, 5, 2, 4, 6, 0, 5, 0, 4, 1, 3, 4, 1, 1, 2, 6, 4, 1, 4, 2, 2, 7, 0, 8, 4, 4, 3, 2, 1, 2

Offset: 1

Author

Labos Elemer, Jun 07 2001

Keywords

• nonn

Examples

			Between 113 and 127 the 6 squarefree numbers are 114, 115, 118, 119, 122, 123, so a(30)=6.
From _Gus Wiseman_, Nov 06 2024: (Start)
The a(n) squarefree numbers for n = 1..16:
  1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16
  ---------------------------------------------------------------
  .   .   6   10  .   14  .   21  26  30  33  38  42  46  51  55
                      15      22          34  39              57
                                          35                  58
(End)
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A005117, A013928.

Cf. A179211. [Reinhard Zumkeller, Jul 05 2010]

Counting all composite numbers (not just squarefree) gives A046933.

The version for nonsquarefree numbers is A061399.

Zeros are A068360.

The version for prime-powers is A080101.

Partial sums are A337030.

The version for non-prime-powers is A368748.

Excluding prime(n+1) from the range gives A373198.

Ones are A377430.

Positives are A377431.

The version for perfect-powers is A377432.

The version for non-perfect-powers is A377433 + 2.

For squarefree numbers (A005117) between primes:

- length is A061398 (this sequence)

- min is A112926

- max is A112925

- sum is A373197

For squarefree numbers between powers of two:

- length is A077643 (except initial terms), partial sums A143658

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

- max is A372889, difference A373126

- sum is A373123

For primes between powers of two:

- length is A036378

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

- max is A014234, difference A013603

- sum is A293697 (except initial terms)

Cf. A001223, A049093, A049094, A076259, A077641, A377434, A377436.

Programs

Maple

p:= 2:
for n from 1 to 200 do
  q:= nextprime(p);
A[n]:= nops(select(numtheory:-issqrfree, [$p+1..q-1]));
p:= q;
od:
seq(A[i],i=1..200); # Robert Israel, Jan 06 2017

Mathematica

a[n_] := Count[Range[Prime[n]+1, Prime[n+1]-1], _?SquareFreeQ];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)
Count[Range[#[[1]]+1,#[[2]]-1],?(SquareFreeQ[#]&)]&/@Partition[ Prime[ Range[120]],2,1] (* _Harvey P. Dale, Oct 14 2021 *)

PARI

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=issquarefree(i)); write("b061398.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

PARI

a(n) = my(pp=prime(n)+1); sum(k=pp, nextprime(pp)-1, issquarefree(k)); \\ Michel Marcus, Feb 28 2019

Python

from math import isqrt
from sympy import mobius, prime, nextprime
def A061398(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))-1 # Chai Wah Wu, Jun 01 2024

Formula

a(n) = A013928(A000040(n+1)) - A013928(A000040(n)) - 1. - Robert Israel, Jan 06 2017

a(n) = A373198(n) - 1. - Gus Wiseman, Nov 06 2024