cp's OEIS Frontend

A097906 Decimal expansion of Sum_{i >= 1} i/prime(i)^2.

%I A097906 #46 Feb 03 2018 20:46:22
%S A097906 1,1,4,9,0,6,4,1,7
%N A097906 Decimal expansion of Sum_{i >= 1} i/prime(i)^2.
%C A097906 By the prime number theorem the growth of n/prime(n)^2 is of the same order as 1/(n*log(n)^2). So the convergence of this series is as slow as the convergence of A115563. The current vague estimate is derived solely from the brute force accumulation of the partial sums (see Examples).
%C A097906 For the first 10000000 primes the sum = 1.0949524507.
%C A097906 For the first 20000000 primes the sum = 1.0970205319.
%C A097906 For the first 30000000 primes the sum = 1.0981565164.
%C A097906 For the first 40000000 primes the sum = 1.0989320390.
%C A097906 For the first 50000000 primes the sum = 1.0995170340.
%C A097906 For the first 60000000 primes the sum = 1.0999846910.
%C A097906 For the first 70000000 primes the sum = 1.1003730572.
%C A097906 For the first 80000000 primes the sum = 1.1007044057.
%C A097906 For the first 90000000 primes the sum = 1.1009928516.
%C A097906 For the first 100000000 primes the sum = 1.1012478922.
%C A097906 However, this does not take into account the non-negligible contribution from the tail.
%C A097906 From _Charles R Greathouse IV_, Dec 04 2014: (Start)
%C A097906 My current estimate of the constant is 1.1490642... based on a summation of the primes up to 10^10, 1.10463086777...
%C A097906 plus the integral from pi(10^10) + 1/2 to infinity of x/ali(x)^2 where ali(x) is the inverse logarithmic integral, which contributes 0.04443338335... (End)
%e A097906 1.14906417...
%Y A097906 Cf. A115563, A085548.
%K A097906 cons,more,nonn
%O A097906 1,3
%A A097906 _Pierre CAMI_, Sep 04 2004, Dec 04 2014
%E A097906 Corrected and extended by _Charles R Greathouse IV_, Dec 04 2014
%E A097906 Extended by _David Broadhurst_, Dec 07 2014