OFFSET
0,3
COMMENTS
Partial sums of A000196. - Michel Marcus, Mar 01 2016
It seems that 197 is the largest prime in this sequence. Tested for n <= 10^11. - Hugo Pfoertner, Oct 26 2020
By drawing a picture of the sum Integral_{x=0..n} ceiling(sqrt(x)) dx, one easily sees that it is equal to n*m - Sum_{k=1..m} (k^2 - 1) with m = floor(sqrt(n)), whence the formula. - M. F. Hasler, Apr 23 2022
We can prove that 197 is the largest prime in this sequence. From the formula, 6*a(n) = m*(6n - 2m^2 - 3m + 5) where m = floor(sqrt(n)). Therefore 6*a(n) is divisible by m, which means that a(n) is divisible by A060789(m). For n>48, we have m>6, so A060789(m)>1, so a(n) is not prime; testing all n up to 48, we see that a(47)=197 is the last prime. - Mikhail Lavrov, Dec 09 2023
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd Edition, Addison-Wesley, 1994, Eq. 3.27 on page 87.
D. E. Knuth, The Art of Computer Programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 43 of section 1.2.4.
K. H. Rosen, Discrete Mathematics and Its Application, 6th Edition, McGraw-Hill, 2007, Ex. 25 of section 2.4.
LINKS
David A. Corneth, Table of n, a(n) for n = 0..9999 (first 1001 terms from G. C. Greubel)
M. Griffiths, More sums involving the floor function, Math. Gaz., 86 (2002), 285-287.
Michael Penn, Wringing out one more result., YouTube video, 2021.
FORMULA
a(0)=0, a(1)=1; a(n) = 2*a(n-1) - a(n-2) if n is not a perfect square; a(n) = 2*a(n-1) - a(n-2) + 1 if n is a perfect square.
a(n) = floor(sqrt(n)) * (n-1/6*(2*floor(sqrt(n))+5)*(floor(sqrt(n))-1)). - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
a(n) = (2/3)*n^(3/2) - (1/2)*n + O(sqrt(n)). - Charles R Greathouse IV, Jan 12 2012
G.f.: Sum_{k>=1} x^(k^2)/(1 - x)^2. - Ilya Gutkovskiy, Dec 22 2016
a(n) = m*(n - m^2/3 - m/2 + 5/6) where m = floor(sqrt(n)). - M. F. Hasler, Apr 23 2022
EXAMPLE
G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 11*x^7 + 13*x^8 + 16*x^9 + ...
MAPLE
seq(add(floor(sqrt(k)), k=0..n), n=0..59);
MATHEMATICA
Accumulate[Floor[Sqrt[Range[0, 60]]]] (* Harvey P. Dale, Feb 16 2011 *)
Table[Sum[Floor[Sqrt[i]], {i, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 22 2016 *)
PROG
(PARI) a(n)=sum(k=1, n, sqrtint(k)) \\ Charles R Greathouse IV, Jan 12 2012
(PARI) a(n)=my(k=sqrtint(n)); k*(n-(2*k+5)/6*(k-1)) \\ Charles R Greathouse IV, Jan 12 2012
(Magma) [&+[Floor(Sqrt(k)): k in [0..n]]: n in [0..50]]; // G. C. Greubel, Feb 26 2018
(Python)
from math import isqrt
def A022554(n): return (m:=isqrt(n))*(m*(-(m<<1)-3)+6*n+5)//6 # Chai Wah Wu, Aug 03 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michel Tixier (tixier(AT)dyadel.net)
EXTENSIONS
More terms from Yong Kong (ykong(AT)curagen.com), Mar 10 2001
STATUS
approved