A025224 a(n) = floor(Sum_{k=1..n} sqrt(k)).

0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 29, 32, 36, 40, 44, 48, 52, 57, 61, 66, 70, 75, 80, 85, 90, 95, 101, 106, 112, 117, 123, 129, 134, 140, 146, 152, 159, 165, 171, 178, 184, 191, 197, 204, 211, 218, 224, 231, 239, 246, 253, 260, 268, 275, 282, 290, 298, 305, 313, 321, 329, 337

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

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.

Crossrefs

Cf. A022554, A352077.

Programs

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 *)

PARI

a(n) = floor(sum(k=0, n, sqrt(k))); \\ Michel Marcus, Mar 01 2016

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