A363499 a(n) = Sum_{k=0..n} floor(sqrt(k))^5.

0, 1, 2, 3, 35, 67, 99, 131, 163, 406, 649, 892, 1135, 1378, 1621, 1864, 2888, 3912, 4936, 5960, 6984, 8008, 9032, 10056, 11080, 14205, 17330, 20455, 23580, 26705, 29830, 32955, 36080, 39205, 42330, 45455, 53231, 61007, 68783, 76559, 84335, 92111, 99887

Offset: 0

Author

Hans J. H. Tuenter, Jun 05 2023

Keywords

• nonn
• easy

Comments

Partial sums of the fifth powers of the terms of A000196.

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000

Crossrefs

Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), A363498 (4th), this sequence (5th).

Programs

Mathematica

Table[(n + 1) #^5 - (1/84) # (# + 1)*(2 # + 1)*(3 # - 1)*(10 #^3 - 7 # + 4) &[Floor@ Sqrt[n]], {n, 0, 42}] (* Michael De Vlieger, Jun 10 2023 *)

Python

from math import isqrt
def A363499(n): return (m:=isqrt(n))**5 *(n+1) - (m*(m+1)*(2*m+1)*(3*m-1)*(10*m**3-7*m+4))//84 # Karl-Heinz Hofmann, Jul 17 2023

Formula

a(n) = (n+1)*m^5 - (1/84)*m*(m+1)*(2*m+1)*(3*m-1)*(10*m^3-7*m+4), where m = floor(sqrt(n)).