A051201 - OEIS

%I #29 Nov 19 2024 00:47:40

%S 1,3,4,7,8,12,13,15,19,21,22,28,29,31,33,39,40,43,44,51,53,55,56,60,

%T 66,68,70,73,74,83,84,87,89,91,93,103,104,106,108,112,113,123,124,127,

%U 130,132,133,138,146,149,151,154,155,159,161,172,174,176,177,183,184,186

%N Sum of elements of the set { [ n/k ] : 1 <= k <= n }.

%H Alois P. Heinz, <a href="/A051201/b051201.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = m*(m+1)/2 + Sum_{k=1..floor(n/(m+1))} floor(n/k), where m is the largest number such that m*(m+1) <= n, i.e., m=floor((sqrt(4*n+1)-1)/2 ). - _Max Alekseyev_, Feb 12 2012

%F From _Ridouane Oudra_, Nov 02 2024: (Start)

%F a(n) = r*(r-1)/2 + Sum_{k=1..floor(sqrt(n))} floor(n/k), where r = floor(sqrt(n+1) + 1/2).

%F a(n) = (1/4)*t*(t-1) + (1/2)*Sum_{k=1..n} floor(n/k), where t = floor(sqrt(4*n + 1)).

%F a(n) = (1/4)*A000267(n)*A055086(n) + (1/2)*A006218(n). (End)

%p seq(add(j, j in {seq(floor(n/i), i=1..n)}), n=1..100); # _Ridouane Oudra_, Nov 02 2024

%t a[n_] := With[{m = Quotient[Floor@Sqrt[4n+1]-1, 2]}, m(m+1)/2 + Sum[ Quotient[n, k], {k, 1, Quotient[n, m+1]}]];

%t Array[a, 100] (* _Jean-François Alcover_, Nov 20 2020, after _Max Alekseyev_ *)

%o (PARI) { a(n) = m=(sqrtint(4*n+1)-1)\2; m*(m+1)/2 + sum(k=1,n\(m+1),n\k) } \\ _Max Alekseyev_, Feb 12 2012

%o (Python)

%o from math import isqrt

%o def A051201(n): return ((m:=isqrt((n<<2)+1)+1>>1)*(m-1)>>1)+sum(n//k for k in range(1,n//m+1)) # _Chai Wah Wu_, Oct 31 2023

%Y Cf. A006218, A000267, A055086.

%K nonn,changed

%O 1,2

%A _David W. Wilson_