A338434 Sum of the numbers less than or equal to sqrt(n) whose square does not divide n.

0, 0, 0, 0, 2, 2, 2, 0, 2, 5, 5, 3, 5, 5, 5, 3, 9, 6, 9, 7, 9, 9, 9, 7, 9, 14, 11, 12, 14, 14, 14, 8, 14, 14, 14, 9, 20, 20, 20, 18, 20, 20, 20, 18, 17, 20, 20, 14, 20, 22, 27, 25, 27, 24, 27, 25, 27, 27, 27, 25, 27, 27, 24, 21, 35, 35, 35, 33, 35, 35, 35, 24, 35, 35, 30, 33, 35, 35, 35, 29, 32, 44, 44, 42, 44, 44, 44, 42, 44, 41, 44, 42, 44, 44, 44, 38, 44, 37, 41, 37

Offset

1,5

Comments

Equal to sum of the numbers less than sqrt(n) whose square does not divide n. - Michel Marcus and Chai Wah Wu, Feb 07 2021

Links

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

Formula

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

a(n) = floor(sqrt(n))*(floor(sqrt(n))+1)/2-sigma(sqrt(n/A007913(n))) = A000217(A000196(n))-A000203(sqrt(n/A007913(n))). - Chai Wah Wu, Jan 31 2021

Example

a(9) = 2; floor(sqrt(n)) = 3 and 2^2 does not divide 9, so a(9) = 2.
a(10) = 5; floor(sqrt(n)) = 3 and the squares of 2 and 3 do not divide 10, so a(10) = 2 + 3 = 5.

Mathematica

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

Prog

(PARI) a(n) = sum(k=1, sqrtint(n), if (n % k^2, k)); \\ Michel Marcus, Jan 31 2021
(Python)
from sympy import divisor_sigma, integer_nthroot
from sympy.ntheory.factor_ import core
def A338434(n):
    m = integer_nthroot(n, 2)[0]
    return m*(m+1)//2-divisor_sigma(integer_nthroot(n//core(n, 2), 2)[0]) # Chai Wah Wu, Jan 31 2021

Crossrefs

Cf. A000217, A000196, A000203, A007913, A338228, A338231, A338233, A338234, A338236, A338430.

Sequence in context: A349437 A215594 A230291 * A059288 A319109 A217670

Adjacent sequences: A338431 A338432 A338433 * A338435 A338436 A338437

Keyword

nonn,easy,changed

Author

Wesley Ivan Hurt, Jan 30 2021

Status

approved