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

1,2

Comments

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

Links

Table of n, a(n) for n=1..59.

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)).

Mathematica

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

Prog

(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))

Crossrefs

Cf. A066839, A176314.

Sequence in context: A062837 A190670 A226115 * A073170 A014689 A117206

Adjacent sequences: A359500 A359501 A359502 * A359504 A359505 A359506

Keyword

nonn

Author

Chai Wah Wu, Jan 24 2024

Status

approved