A378086 Number of nonsquarefree numbers < prime(n).

0, 0, 1, 1, 3, 4, 5, 6, 7, 11, 11, 13, 14, 14, 16, 20, 22, 23, 25, 26, 27, 29, 31, 33, 36, 39, 39, 40, 41, 42, 49, 50, 53, 53, 57, 58, 61, 63, 64, 68, 70, 71, 74, 75, 76, 77, 81, 84, 86, 87, 88, 90, 91, 97, 99, 101, 103, 104, 107, 109, 109, 113, 119, 120, 121

Offset

1,5

Links

Table of n, a(n) for n=1..65.

Formula

a(n) = A057627(prime(n)).

Example

The nonsquarefree numbers counted under each term begin:
  n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9: n=10: n=11: n=12:
  --------------------------------------------------------------
   .    .    4    4    9    12   16   18   20   28    28    36
                       8    9    12   16   18   27    27    32
                       4    8    9    12   16   25    25    28
                            4    8    9    12   24    24    27
                                 4    8    9    20    20    25
                                      4    8    18    18    24
                                           4    16    16    20
                                                12    12    18
                                                9     9     16
                                                8     8     12
                                                4     4     9
                                                            8
                                                            4

Mathematica

Table[Length[Select[Range[Prime[n]], !SquareFreeQ[#]&]], {n, 100}]

Prog

(Python)
from math import isqrt
from sympy import prime, mobius
def A378086(n): return (p:=prime(n))-sum(mobius(k)*(p//k**2) for k in range(1, isqrt(p)+1)) # Chai Wah Wu, Dec 05 2024

Crossrefs

For nonprime numbers we have A014689.

Restriction of A057627 to the primes.

First-differences are A061399 (zeros A068361), squarefree A061398 (zeros A068360).

For composite instead of squarefree we have A065890.

For squarefree we have A071403, differences A373198.

Greatest is A378032 (differences A378034), restriction of A378033 (differences A378036).

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.

A070321 gives the greatest squarefree number up to n.

A112925 gives the greatest squarefree number between primes, differences A378038.

A112926 gives the least squarefree number between primes, differences A378037.

A120327 gives the least nonsquarefree number >= n, first-differences A378039.

A377783 gives the least nonsquarefree > prime(n), differences A377784.

Cf. A013928, A046933, A053797, A053806, A072284, A076259, A224363, A337030, A377049, A378040, A378082, A378084.

Sequence in context: A163078 A050034 A039056 * A326754 A047562 A354270

Adjacent sequences: A378083 A378084 A378085 * A378087 A378088 A378089

Keyword

nonn

Author

Gus Wiseman, Dec 04 2024

Status

approved