A165453 Linear interpolation of the sequence that maps an entry of A002378 to the corresponding entry of A006331.

0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 288, 296, 304

Offset: 0

Author

Friedrich Regen (friedrich.regen(AT)tu-ilmenau.de), Sep 20 2009

Keywords

• easy
• nonn

Crossrefs

Cf. A002378, A006331.

Partial sums of A000194.

Programs

Mathematica

f[n_]:=Round[Sqrt[n]]; a=0;lst={};Do[AppendTo[lst,a+=f[n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
Table[Sum[Floor[Sqrt[n + 1 - k] + 1/2], {k, n + 1}], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 01 2020 *)

Python

from math import isqrt
def A165453(n): return (k:=(m:=isqrt(n))+(n-m*(m+1)>=1))*(3*n+1-k**2)//3 # Chai Wah Wu, Jun 19 2024

Formula

a(0)=0, a(n) = max(min(a(n-1)+x, n+a(n-x))), where the maximum is taken over all values for x from 1 to n.

a(n) = Sum_{k=1..n} floor(sqrt(k)+1/2). - Wesley Ivan Hurt, Dec 01 2020

a(n) = (1/3)*t*(3*n + 1 - t^2), where t = floor(sqrt(n)+1/2). - Ridouane Oudra, Feb 22 2021