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%I A306834 #17 Apr 07 2019 17:26:48
%S A306834 1,8,23,3,53,184,303,65,331,952,1293,1737,1135,2872,3577,1475,1357,
%T A306834 6526,7799,3073,1344,12490,14399,16535,948,502,24367,9121,7631,33914,
%U A306834 37851,42043,1663,51290,56505,20647,33875,73944,80457,87377,47358,34106,1033,119023,31972,137042,146959,157663
%N A306834 Numerator of the barycenter of first n primes defined as a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
%C A306834 It appears that lim_{n->infinity} (1/n)*(A014285(n)/A007504(n)) = k, where k is a constant around 2/3.
%H A306834 Robert Israel, <a href="/A306834/b306834.txt">Table of n, a(n) for n = 1..10000</a>
%F A306834 a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).
%F A306834 a(n) = numerator(A014285(n)/A007504(n)).
%p A306834 N:= 100: # for a(1)..a(N)
%p A306834 Primes:= map(ithprime, [$1..N]):
%p A306834 S1:= ListTools:-PartialSums(Primes):
%p A306834 S2:= ListTools:-PartialSums(zip(`*`,Primes, [$1..N])):
%p A306834 map(numer,zip(`/`,S2,S1)); # _Robert Israel_, Apr 07 2019
%t A306834 a[n_]:=Sum[i*Prime[i],{i,1,n}]/Sum[Prime[i],{i,1,n}];
%t A306834 Table[a[n]//Numerator,{n,1,40}]
%o A306834 (PARI) a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ _Michel Marcus_, Mar 15 2019
%Y A306834 Cf. A272206, A007504, A014285.
%K A306834 nonn,frac,look
%O A306834 1,2
%A A306834 _Andres Cicuttin_, Mar 12 2019

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