A061399 Number of nonsquarefree integers between primes prime(n) and prime(n+1).

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

Offset: 1

Author

Labos Elemer, Jun 07 2001

Keywords

• nonn

Examples

			Between 113 and 127 the 7 numbers which are not squarefree are {116,117,120,121,124,125,126}, so a(30)=7.
From _Gus Wiseman_, Dec 07 2024: (Start)
The a(n) nonsquarefree numbers for n = 1..15:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ----------------------------------------------------------
   .   4   .   8  12  16  18  20  24   .  32  40   .  44  48
               9                  25      36          45  49
                                  27                      50
                                  28                      52
(End)
		

Links

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

Crossrefs

Zeros are A068361.

First differences of A378086, restriction of A057627 to the primes.

Other classes (instead of nonsquarefree):

- For composite we have A046933, first differences of A065890.

- For squarefree see A061398, A068360, A071403, A373197, A373198, A377431.

- For prime power we have A080101.

- For non prime power we have A368748, see A378616.

- For perfect power we have A377432, zeros A377436.

- For non perfect power we have A377433, A029707.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers, differences A076259.

A013929 lists the nonsquarefree numbers, differences A078147.

A120327 gives the least nonsquarefree number >= n.

Cf. A007674, A053806, A072284, A073247, A179211, A224363.

Cf. A013928, A067535, A070321, A112925, A112926, A240473, A240474.

Cf. A045882, A377049, A377783, A378032, A378618.

Programs

Mathematica

Count[Range[#[[1]]+1,#[[2]]-1],?(!SquareFreeQ[#]&)]&/@Partition[Prime[Range[120]],2,1] (* _Harvey P. Dale, Mar 31 2024 *)

PARI

{ n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=!issquarefree(i)); write("b061399.txt", n++, " ", a); q=p ) } \\ Harry J. Smith, Jul 22 2009

PARI

a(n) = my(p=prime(n)); sum(k=p, nextprime(p+1), ! issquarefree(k)); \\ Michel Marcus, Dec 09 2024

Python

from sympy import mobius, prime
def A061399(n): return sum(not mobius(m) for m in range(prime(n)+1,prime(n+1))) # Chai Wah Wu, Jul 20 2024