OFFSET
0,3
COMMENTS
Shekatkar took Ramanujan's formula for sum of the square roots of first n natural numbers, and generalized to include r-th roots where r is any real number greater than 1, using simple properties of Riemann integrable functions. - Jonathan Vos Post, Apr 05 2012
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. Ramanujan, On the sum of the square roots of the first n natural numbers, Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
S. Ramanujan, On the sum of the square roots of the first n natural numbers; Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
Snehal Shekatkar, On the sum of the r'th roots of first n natural numbers, arXiv:1204.0877 [math.NT], 2012-2013.
FORMULA
a(n) ~= floor ((4n + 3)sqrt(n)/6 - exp(-Pi / 2)). - Charles R Greathouse IV, Jul 29 2007. Corrected by Carl R. White, Jan 22 2009
MATHEMATICA
s=0; lst={}; Do[s+=Sqrt[n]; AppendTo[lst, c=Floor[s]], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
Floor@HarmonicNumber[Range[0, 30], -1/2] (* Vladimir Reshetnikov, Nov 07 2015 *)
Floor[Accumulate[Sqrt[Range[0, 70]]]] (* Harvey P. Dale, Apr 23 2022 *)
PROG
(PARI) a(n) = floor(sum(k=0, n, sqrt(k))); \\ Michel Marcus, Mar 01 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved