Terms listed are the only three primes p found to satisfy the condition that p = prime(m) = Sum_{i=1..k} prime(i) where m=prime(k).
Let S(k) be the sum of the first k primes, and let PP(k) = prime(prime(k)); then the terms of the sequence are the values of prime(prime(k)) at those values of k at which S(k) = PP(k). (This occurs at k = 2, 4 and 6.)
Given the behavior of the ratio S(k)/PP(k) over the range of values of k shown in the table below, it seems very unlikely that this ratio will return to 1 for any k beyond the values that have been tested, and thus very likely that a(3) = 41 = PP(6) is the final term in the sequence:
k S(k) PP(k) S(k)/PP(k)
====== =========== ======== ==============
1 2 3 0.666666...
2 5 = 5 1
3 10 11 0.909090...
4 17 = 17 1
5 28 31 0.903225...
6 41 = 41 1
7 58 59 0.983050...
8 77 67 1.149253...
9 100 83 1.204819...
10 129 109 1.183486...
...
100 24133 3911 6.170544...
1000 3682913 80917 45.514700...
10000 496165411 1366661 363.049367...
100000 62260698721 20491057 3038.432752... (End)
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