A306834 - OEIS

%I #17 Apr 07 2019 17:26:48

%S 1,8,23,3,53,184,303,65,331,952,1293,1737,1135,2872,3577,1475,1357,

%T 6526,7799,3073,1344,12490,14399,16535,948,502,24367,9121,7631,33914,

%U 37851,42043,1663,51290,56505,20647,33875,73944,80457,87377,47358,34106,1033,119023,31972,137042,146959,157663

%N 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 It appears that lim_{n->infinity} (1/n)*(A014285(n)/A007504(n)) = k, where k is a constant around 2/3.

%H Robert Israel, <a href="/A306834/b306834.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = numerator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).

%F a(n) = numerator(A014285(n)/A007504(n)).

%p N:= 100: # for a(1)..a(N)

%p Primes:= map(ithprime, [$1..N]):

%p S1:= ListTools:-PartialSums(Primes):

%p S2:= ListTools:-PartialSums(zip(`*`,Primes, [$1..N])):

%p map(numer,zip(`/`,S2,S1)); # _Robert Israel_, Apr 07 2019

%t a[n_]:=Sum[i*Prime[i],{i,1,n}]/Sum[Prime[i],{i,1,n}];

%t Table[a[n]//Numerator,{n,1,40}]

%o (PARI) a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ _Michel Marcus_, Mar 15 2019

%Y Cf. A272206, A007504, A014285.

%K nonn,frac,look

%O 1,2

%A _Andres Cicuttin_, Mar 12 2019