A359503 Partial sums of A066839.

1, 2, 3, 6, 7, 10, 11, 14, 18, 21, 22, 28, 29, 32, 36, 43, 44, 50, 51, 58, 62, 65, 66, 76, 82, 85, 89, 96, 97, 108, 109, 116, 120, 123, 129, 145, 146, 149, 153, 165, 166, 178, 179, 186, 195, 198, 199, 215, 223, 231, 235, 242, 243, 255, 261, 275, 279, 282, 283

Offset: 1

Author

Chai Wah Wu, Jan 24 2024

Keywords

• nonn

Comments

a(n) is the sum of all divisors d of k such that d^2 <= k where k ranges from 1 to n.

Crossrefs

Cf. A066839, A176314.

Programs

Mathematica

Table[Select[Divisors[n], # <= Sqrt[n]&]//Total, {n, 1, 60}]//Accumulate (* Jean-François Alcover, Jan 26 2024 *)

Python

from itertools import takewhile
from sympy import divisors
def A359503(n): return sum(sum(takewhile(lambda x:x**2<=i,divisors(i))) for i in range(1,n+1))

Formula

a(n) = m*(6*n+5-m*(2*m+3))/6 + Sum_{k=1..n, i=1..floor(sqrt(k))} [(k-1) mod i] - [k mod i] where m = floor(sqrt(n)).

a(n) = m*(6*n+5-m*(2*m+3))/6 + Sum_{k=1..n, i=1..floor(sqrt(k))} (k-1) mod i - Sum_{k=1..n} A176314(k) where m = floor(sqrt(n)).