How can I express a sum indexed by primes in Mathematica? Two examples that I am interested in are
(1) where the primes go from $p=2$ to, say, $p=17$.
(2) It would also be useful to have the sum go from $p=2$ to largest prime less than $n$. I know the latter can be estimated using the prime number theorem.
Here are two possibilities using the fact that p = Prime[i] is the $i^{th}$ prime. A simple sum indexed by primes from $p=2$ to $p=17$ (the $7^{th}$ prime):
Sum[f[Prime[i]], {i, 7}]
(* f[2] + f[3] + f[5] + f[7] + f[11] + f[13] + f[17] *)
And a sum from $p = 2$ to the largest prime less than $n$:
Block[{n = 18},
Sum[f[Prime[i]], {i, PrimePi[n]}]
]
(* f[2] + f[3] + f[5] + f[7] + f[11] + f[13] + f[17] *)
You may also find NextPrime[n, k] useful -- it finds the $k^{th}$ above $n$ (or below for negative $k$).
Consider
sumOverPrimes[f_, n_Integer /; n > 1] :=
Total @ Table[f[Prime[i]], {i, 1, PrimePi[n]}]
With this function,
sumOverPrimes[f, 18]
f[2] + f[3] + f[5] + f[7] + f[11] + f[13] + f[17]
and
1 + sumOverPrimes[#^2 + 1 &, 6]
42
