A359503 - OEIS

%I #33 Jan 26 2024 06:56:08

%S 1,2,3,6,7,10,11,14,18,21,22,28,29,32,36,43,44,50,51,58,62,65,66,76,

%T 82,85,89,96,97,108,109,116,120,123,129,145,146,149,153,165,166,178,

%U 179,186,195,198,199,215,223,231,235,242,243,255,261,275,279,282,283

%N Partial sums of A066839.

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

%F 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)).

%F 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)).

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

%o (Python)

%o from itertools import takewhile

%o from sympy import divisors

%o def A359503(n): return sum(sum(takewhile(lambda x:x**2<=i,divisors(i))) for i in range(1,n+1))

%Y Cf. A066839, A176314.

%K nonn

%O 1,2

%A _Chai Wah Wu_, Jan 24 2024