From Jon E. Schoenfield, Jan 14 2019:

Define the partial sum s(j) = Sum_{k=1..j} 1/(k*prime(k)), let S = lim_{j->inf} s(j) be the sum whose decimal expansion gives the terms of this sequence, and let g(j_1,j_2) = s(j_2) - s(j_1); then g(j_1,j_2) is the contribution of the primes in the half-open interval (p1=prime(j_1), p2=prime(j_2)] to the sum S. 

If the "density" of primes in the vicinity of x is approximately f(x) = 1/log(x) (see the Prime Number Theorem link), then given any two primes p_1 < p_2, the number of primes in (p_1, p_2] should be about 

     Integral_{x=p_1..p_2} f(x)*dx 

   = Integral_{x=p_1..p_2} (1/log(x))*dx 

and g(j_1,j_2) should be approximately 

   h(p_1,p_2) = Integral_{x=p_1..p_2} (1/(y(x)*x*log(x)))*dx 

where y(x) and x in the integral correspond to k and prime(k) in the summation, respectively; i.e., y(x) is a continuous function approximating primepi(x) and can be computed as 

   y(x) = primepi(p_1) + Integral_{x=p_1..x} (1/log(x))*dx. 

If we have the value of j_1 = primepi(p_1) but not that of j_2 = primepi(p_2), we can use numerical integration to evaluate 

   y(x) = j1 + Integral_{x=p1..x} (1/log(x))*dx 

and 

   h(p_1,p_2) = Integral_{x=p_1..p_2} (1/(y(x)*x*log(x)))*dx. 

However, if we know the prime indices of both p_1 and p_2, i.e., if we know both j_1 and j_2, then we can define for the interval (p_1,p_2] a correction factor 

   c(p_1,p_2) = (j_2 - j_1)/Integral_{x=p_1..p_2} (1/log(x))*dx 

(i.e., c(p_1,p_2) is the ratio of the actual number of primes in (p_1,p_2] to the number of primes predicted to occur in that interval using 1/log(x) as the density of the primes) so that 

   j_1 + c(p_1,p_2)*Integral_{x=p_1..p_2} (1/log(x))*dx = j_2 

and, in lieu of the function y(x), we can use the improved function 

   y'(x) = j1 + c(p_1,p_2)*Integral_{x=p_1..x} (1/log(x))*dx 

and, in lieu of h(p_1,p_2), we can use 

   h'(p_1,p_2) = c(p_1,p_2)*Integral_{x=p_1..p_2} (1/(x*y'(x)*log(x)))*dx 

as an improved estimate for g(j_1,j_2). 

Thus, given an actual partial sum s(j_0) = Sum_{k=1..j_0} 1/(k*prime(k)) and a set of M pairs (j_m, prime(j_m)) for m = 1..M, with j_0 < j_1 < j_2 < ... < j_M, we can approximate the "tail" of the summation beyond prime(j_0) as 

   t(prime(j_0)) = 

              (Sum_{m=1..M} h'(prime(j_{m-1}), prime(j_m))) 

             + lim_{x->inf} h(prime(j_M), x). 

(See the Example section.)