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A300287
a(n) = floor((1/n) * Sum_{k=1..n} sqrt(k)).
0
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
OFFSET
1,8
LINKS
Mircea Merca, On the Arithmetic Mean of the Square Roots of the First n Positive Integers, The College Mathematics Journal, Vol. 48, No. 2 (March 2017), pp. 129-133.
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.
Thomas P. Wihler, Rounding the arithmetic mean value of the square roots of the first n integers, arXiv:1803.00362 [math.NT], 2018.
FORMULA
a(n) = floor((2/3)*sqrt(n+1)*(1+1/(4*n))). See Theorem 1 of Wihler paper.
MATHEMATICA
Table[Floor[1/n Sum[Sqrt[k], {k, n}]], {n, 200}] (* Vincenzo Librandi, Mar 02 2018 *)
PROG
(PARI) a(n) = floor(sum(k=1, n, sqrt(k))/n);
(Magma) [Floor(&+[Sqrt(k)/n: k in [1..n]]): n in [1..100]]; // Bruno Berselli, Mar 02 2018
CROSSREFS
Cf. A025224.
Sequence in context: A263206 A230501 A287272 * A226764 A344420 A206244
KEYWORD
nonn
AUTHOR
Michel Marcus, Mar 02 2018
STATUS
approved