A338234 Sum of the numbers less than n whose square does not divide n.

0, 0, 2, 3, 9, 14, 20, 25, 32, 44, 54, 63, 77, 90, 104, 113, 135, 149, 170, 187, 209, 230, 252, 273, 294, 324, 347, 375, 405, 434, 464, 489, 527, 560, 594, 618, 665, 702, 740, 777, 819, 860, 902, 943, 986, 1034, 1080, 1121, 1168, 1219, 1274, 1323, 1377, 1427, 1484, 1537

Offset

1,3

Links

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

Formula

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

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

Example

a(7) = 20; 1^2|7, but the squares of 2,3,4,5 and 6 do not. So a(7) = 2 + 3 + 4 + 5 + 6 = 20.
a(8) = 25; 1^2|8 and 2^2|8, but the squares of 3,4,5,6 and 7 do not. So a(8) = 3 + 4 + 5 + 6 + 7 = 25.

Mathematica

Table[Sum[k*(Ceiling[n/k^2] - Floor[n/k^2]), {k, n - 1}], {n, 60}]

Prog

(PARI) a(n) = sum(k=1, n-1, if (n % k^2, k)); \\ Michel Marcus, Jan 31 2021
(PARI) a(n) = my(res = binomial(n, 2), f = factor(n)); f[, 2]>>=1; res-sigma(factorback(f))+(n==1) \\ David A. Corneth, Jan 31 2021
(Python)
from sympy import divisor_sigma, integer_nthroot
from sympy.ntheory.factor_ import core
def A338234(n):
    return 0 if n <= 1 else n*(n-1)//2 - divisor_sigma(integer_nthroot(n//core(n, 2), 2)[0]) # Chai Wah Wu, Jan 31 2021

Crossrefs

Cf. A000217, A000203, A007913, A338228, A338231, A338233, A338236, A338430, A338434.

Sequence in context: A057293 A109662 A116222 * A048038 A113501 A101067

Adjacent sequences: A338231 A338232 A338233 * A338235 A338236 A338237

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 30 2021

Status

approved