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URL: https://oeis.org/A359503

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A359503
Partial sums of A066839.
1
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.
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
Sequence in context: A062837 A190670 A226115 * A073170 A014689 A117206
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Jan 24 2024
STATUS
approved