A307716 Denominator of the barycenter of first n primes defined as a(n) = denominator(Sum_{i=1..n} (i*prime(i)) / Sum_{i=1..n} prime(i)).

1, 5, 10, 1, 14, 41, 58, 11, 50, 129, 160, 197, 119, 281, 328, 127, 110, 501, 568, 213, 89, 791, 874, 963, 53, 27, 1264, 457, 370, 1593, 1720, 1851, 71, 2127, 2276, 809, 1292, 2747, 2914, 3087, 1633, 1149, 34, 3831, 1007, 4227, 4438, 4661

Offset: 1

Author

Andres Cicuttin, Apr 25 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.

a(n) = A007504(n) if and only if n is in A307414. - Robert Israel, Jul 08 2019

Links

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

Crossrefs

Cf. A306834 (numerators), A272206, A007504, A014285, A307414.

Programs

Maple

S1:= 0:S2:= 0:
for n from 1 to 100 do
  p:= ithprime(n);
  S1:= S1 + p;
  S2:= S2 + n*p;
  A[n]:= denom(S2/S1)
od:
seq(A[i],i=1..100); # Robert Israel, Jul 08 2019

Mathematica

a[n_]:=Sum[i*Prime[i], {i, 1, n}]/Sum[Prime[i], {i, 1, n}];
Table[a[n]//Denominator, {n, 1, 48}]

PARI

a(n) = my(vp=primes(n)); denominator(sum(i=1, n, i*vp[i])/sum(i=1, n, vp[i])) \\ Michel Marcus, Apr 25 2019

Formula

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

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