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%I #50 Feb 18 2024 02:04:00
%S 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,
%T 85,90,95,101,106,112,117,123,129,134,140,146,152,159,165,171,178,184,
%U 191,197,204,211,218,224,231,239,246,253,260,268,275,282,290,298,305,313,321,329,337
%N a(n) = floor(Sum_{k=1..n} sqrt(k)).
%C 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
%H Harvey P. Dale, <a href="/A025224/b025224.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper9/page1.htm">On the sum of the square roots of the first n natural numbers</a>, Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram09.pdf">On the sum of the square roots of the first n natural numbers</a>; Journal of the Indian Mathematical Society, VIII (1915), pp. 173-175.
%H Snehal Shekatkar, <a href="http://arxiv.org/abs/1204.0877">On the sum of the r'th roots of first n natural numbers</a>, arXiv:1204.0877 [math.NT], 2012-2013.
%F 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
%t s=0;lst={};Do[s+=Sqrt[n];AppendTo[lst,c=Floor[s]],{n,0,6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2010 *)
%t Floor@HarmonicNumber[Range[0, 30], -1/2] (* _Vladimir Reshetnikov_, Nov 07 2015 *)
%t Floor[Accumulate[Sqrt[Range[0,70]]]] (* _Harvey P. Dale_, Apr 23 2022 *)
%o (PARI) a(n) = floor(sum(k=0, n, sqrt(k))); \\ _Michel Marcus_, Mar 01 2016
%Y Cf. A022554, A352077.
%K nonn
%O 0,3
%A _Clark Kimberling_
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