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A301277 Nearest integer to mean of first n primes.

%I A301277 #46 Sep 03 2021 09:39:03
%S A301277 2,3,3,4,6,7,8,10,11,13,15,16,18,20,22,24,26,28,30,32,34,36,38,40,42,
%T A301277 45,47,49,51,53,55,58,60,63,65,67,70,72,75,77,80,82,85,87,90,92,94,97,
%U A301277 100,102,105,107,110,113,115,118,121,123,126,128,131,133
%N A301277 Nearest integer to mean of first n primes.
%C A301277 Differs from A075465 where ties are involved. - _R. J. Mathar_, Mar 20 2018
%H A301277 Altug Alkan, <a href="/A301277/b301277.txt">Table of n, a(n) for n = 1..10000</a>
%H A301277 Joel E. Cohen, <a href="http://lab.rockefeller.edu/cohenje/assets/file/415PrimesTwinPrimesTaylorsLawAmStatistician2016.pdf">Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology</a>, The American Statistician, 70 (2016), 399-404.
%F A301277 a(n) = round(A007504(n) / n). - _David A. Corneth_, Mar 22 2018
%F A301277 a(n) ~ prime(n)/2 ~ n*log(n)/2. - _Daniel Forgues_, Mar 22 2018
%e A301277 The means are 2, 5/2, 10/3, 17/4, 28/5, 41/6, 58/7, 77/8, 100/9, 129/10, 160/11, 197/12, 238/13, 281/14, 328/15, 381/16, 440/17, 167/6, 568/19, 639/20, 712/21, 791/22, 38, 321/8, 212/5, ...
%p A301277 m := n -> add(ithprime(j),j=1..n)/n;
%p A301277 m1:=[seq(m(n),n=1..100)];
%p A301277 m2:=map(numer,m1); # A301273
%p A301277 m3:=map(denom,m1); # A301274
%p A301277 m4:=map(round,m1); # A301277
%t A301277 Rest@ FoldList[{Append[First@ #1, #2], If[And[EvenQ@ #1, #2 == 1/2] & @@ {IntegerPart@ #, FractionalPart@ #}, Round@ # + 1, Round@ #] &@ Mean@ First@ #1} &, {{2}, 2}, Prime@ Range[2, 63]][[All, -1]] (* _Michael De Vlieger_, Apr 05 2018 *)
%o A301277 (PARI) a(n) = round(sum(i=1, n, prime(i))/n); \\ _Altug Alkan_, Mar 22 2018
%Y A301277 Mean and variance of primes: A301273/A301274, A301275/A301276, A301277, A273462.
%Y A301277 Cf. A007504, A060620, A062049, A075465, A225804, A250130/A024451.
%K A301277 nonn
%O A301277 1,1
%A A301277 _N. J. A. Sloane_, Mar 18 2018