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A124478 a(n) = prime(n) - floor((2/n)*Sum_{i=1..n} prime(i)). 1


%S -2,-2,-1,-1,0,0,1,0,1,4,2,5,5,3,4,6,8,6,8,8,6,8,7,9,13,12,10,10,7,7,

%T 17,16,17,14,19,17,18,19,18,19,20,17,22,19,18,16,23,29,28,25,24,25,21,

%U 26,27,28,28,25,26,25,22,27,35,34,30,29,37,38,42,38,37,37,40,40,40,39,39,41,40,42

%N a(n) = prime(n) - floor((2/n)*Sum_{i=1..n} prime(i)).

%C Robert Mandl conjectured and Rosser and Schoenfeld proved that prime(n)/2 > (Sum_{i=1..n} prime(i))/n for n >= 9 (implying that a(n) > 0 for n >= 9).

%H Zak Seidov, <a href="/A124478/b124478.txt">Table of n, a(n) for n = 1..1000</a>

%H C. Axler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Axler/axler6.html">On a Sequence involving Prime Numbers</a>, J. Int. Seq. 18 (2015) # 15.7.6

%H Christian Axler, <a href="http://arxiv.org/abs/1606.06874">New bounds for the sum of the first n prime numbers</a>, arXiv:1606.06874 [math.NT], 2016.

%H Pierre Dusart, <a href="http://www.unilim.fr/laco/theses/1998/T1998_01.pdf">Autour de la fonction qui compte le nombre de nombres premiers</a>, Thèse, Université de Limoges, France, (1998), see Section 1.9.

%H M. Hassani, <a href="http://arXiv.org/abs/math/0606765">A Remark on the Mandl's Inequality</a>, arXiv:math/0606765 [math.NT], 2006.

%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0457373-7">Sharper bounds for the Chebyshev functions theta(x) and psi(x)</a>, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.

%t Table[Prime[n] - Floor[(2/n)*Sum[Prime[i], {i, n}]], {n, 100}] (* _Michael De Vlieger_, Jan 31 2015 *)

%K sign

%O 1,1

%A _N. J. A. Sloane_, Dec 17 2006

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Last modified December 6 09:37 EST 2019. Contains 329800 sequences. (Running on oeis4.)