A338228 Number of numbers less than or equal to n whose square does not divide n.

0, 1, 2, 2, 4, 5, 6, 6, 7, 9, 10, 10, 12, 13, 14, 13, 16, 16, 18, 18, 20, 21, 22, 22, 23, 25, 25, 26, 28, 29, 30, 29, 32, 33, 34, 32, 36, 37, 38, 38, 40, 41, 42, 42, 43, 45, 46, 45, 47, 48, 50, 50, 52, 52, 54, 54, 56, 57, 58, 58, 60, 61, 61, 60, 64, 65, 66, 66, 68, 69, 70, 68, 72

Offset

1,3

Links

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Formula

a(n) = n - Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)).

a(n) = n - tau(sqrt(n/A007913(n))) = n - A000005(sqrt(n/A007913(n))). - Chai Wah Wu, Feb 01 2021

a(n) = Sum_{k=1..n} sign(n mod k^2). - Wesley Ivan Hurt, May 09 2021

Example

a(3) = 2; 1^2|3, but 2^2 and 3^2 do not. So a(3) = 2.
a(4) = 2; 1^2|4 and 2^2|4 but 3^2 and 4^2 do not, So a(4) = 2.

Mathematica

Table[Sum[Ceiling[n/k^2] - Floor[n/k^2], {k, n}], {n, 100}]

Prog

(PARI) a(n) = sum(k=1, n, if (n % k^2, 1)); \\ Michel Marcus, Jan 31 2021
(Python)
from sympy import divisor_count, integer_nthroot
from sympy.ntheory.factor_ import core
def A338228(n):
    return n-divisor_count(integer_nthroot(n//core(n, 2), 2)[0]) # Chai Wah Wu, Feb 01 2021

Crossrefs

Cf. A338231, A338233, A338234, A338236, A338430, A338434.

Sequence in context: A130498 A263433 A147806 * A351782 A064574 A059015

Adjacent sequences: A338225 A338226 A338227 * A338229 A338230 A338231

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 30 2021

Status

approved