How to select from a list at specific locations

Question

Suppose that a list can be generated for a sum of primes as follows:

g[n_]:= g[n]= Sum[Prime[j], {j, n}];
lst = Table[g[n], {n, 1, 1000}];

Now that a list has been created what is an efficient way to select values at Lucas number positions ? The result should read {2, 10, 17, 58, ...}. I have tried several version of Select,

Select[Range[10], Extract[ LucasL[#]][ lst[[#]] ] &]
Select[Range[10], Pick[ lst[[#]], LucasL[#] ] &]

and all lead to empty results.

For context: The calculation process above was/is an attempt to speed up the sum

f[n_]:= f[n]= Sum[Prime[i], {i, LucasL[n]}];
Table[f[n], {n,1,50}]

Any ideas on how to select data from particular positions or how to speed up the calculation of the series would be most helpful.

Answer 1

Clear["Global`*"]

g1[n_] := g1[n] = Sum[Prime[j], {j, n}];

t1 = AbsoluteTiming[
   lst1 = Table[g1[n], {n, 1, 1000}];][[1]]

(* 0.142708 *)

The definition of g1 repeatedly calculates identical values of Prime. Using a recursive definition avoids this.

g2[1] = Prime[1];
g2[n_] := g2[n] = g2[n - 1] + Prime[n];

t2 = AbsoluteTiming[
   lst2 = Table[g2[n], {n, 1, 1000}];][[1]]

(* 0.002815 *)

The second method is about 50 times faster

t1/t2

(* 50.6956 *)

The different definitions give identical results.

lst1 === lst2

(* True *)

To find the maximum value of n

nmax = Floor@NMaxValue[{n, LucasL[n] <= 1000}, n]

(* 14 *)

Then, as pointed out in the comments

lst1[[LucasL@Range@nmax]]

(* {2, 10, 17, 58, 160, 501, 1480, 4438, 13101, 38238, 110364, 
    316421, 901478, 2549658} *)