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)).

1, 8, 23, 3, 53, 184, 303, 65, 331, 952, 1293, 1737, 1135, 2872, 3577, 1475, 1357, 6526, 7799, 3073, 1344, 12490, 14399, 16535, 948, 502, 24367, 9121, 7631, 33914, 37851, 42043, 1663, 51290, 56505, 20647, 33875, 73944, 80457, 87377, 47358, 34106, 1033, 119023, 31972, 137042, 146959, 157663

Offset: 1

Author

Andres Cicuttin, Mar 12 2019

Keywords

• nonn
• frac
• look

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A272206, A007504, A014285.

Programs

Maple

N:= 100: # for a(1)..a(N)
Primes:= map(ithprime, [$1..N]):
S1:= ListTools:-PartialSums(Primes):
S2:= ListTools:-PartialSums(zip(`*`,Primes, [$1..N])):
map(numer,zip(`/`,S2,S1)); # Robert Israel, Apr 07 2019

Mathematica

a[n_]:=Sum[i*Prime[i],{i,1,n}]/Sum[Prime[i],{i,1,n}];
Table[a[n]//Numerator,{n,1,40}]

PARI

a(n) = numerator(sum(i=1, n, i*prime(i))/sum(i=1, n, prime(i))); \\ Michel Marcus, Mar 15 2019

Formula

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

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