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A338231
Sum of the numbers less than or equal to n whose square does not divide n.
7
0, 2, 5, 7, 14, 20, 27, 33, 41, 54, 65, 75, 90, 104, 119, 129, 152, 167, 189, 207, 230, 252, 275, 297, 319, 350, 374, 403, 434, 464, 495, 521, 560, 594, 629, 654, 702, 740, 779, 817, 860, 902, 945, 987, 1031, 1080, 1127, 1169, 1217, 1269, 1325, 1375, 1430, 1481, 1539
OFFSET
1,2
LINKS
FORMULA
a(n) = n*(n+1)/2 - Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)) * k.
a(n) = n*(n+1)/2-sigma(sqrt(n/A007913(n)) = A000217(n)-A000203(sqrt(n/A007913(n)). - Chai Wah Wu, Jan 31 2021
EXAMPLE
a(3) = 5; 1^1|3, but 2^2 and 3^2 do not. Then 2 + 3 = 5.
a(4) = 7; 1^2|4 and 2^2|4, but 3^2 and 4^2 do not. Then 3 + 4 = 7.
MATHEMATICA
Table[Sum[k (Ceiling[n/k^2] - Floor[n/k^2]), {k, n}], {n, 80}]
PROG
(PARI) a(n) = sum(k=1, 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 A338231(n):
return n*(n+1)//2-divisor_sigma(integer_nthroot(n//core(n, 2), 2)[0]) # Chai Wah Wu, Jan 31 2021
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jan 30 2021
STATUS
approved