[seqfan] Partial sum of A013916 Numbers n such that the sum of the first n primes is prime

[Jonathan Post]

Is this interesting enough, for the recursive reason below?

Partial sum of A013916  Numbers n such that the sum of the first n
primes is prime.

1, 3, 7, 13, 25, 39, 99, 163, 259, 359, 461, 569, 683, 805, 929 ,
1059, 1191, 1337, 1489, 1647, 1809, 1987, 2179, 2377, 2581, 2787,
2995, 3209, 3425, 3721, 4029, 4355, 5013, 5679, 6021, 6371, 6727,
7085, 7511, 7957, 8415, 8875, 9339, 9819, 10303

This is not interesting as such, but for the recursion, whose 2nd
iteration gives the derived sequence:
Numbers n such that the sum of the first n of the (numbers m such that
the sum of the first m primes is prime) is prime:
2, 3, 4, 8, 10, 11, 12, 13, 15, 19, 22, 23, 24, 28, 46

1+ 2+ 4+ 6+ 12+ 14+ 60+ 64 = 163 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100 = 359 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102 = 461 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108 = 569 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114 = 683 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124 = 929 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152 = 1489 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152+ 158 + 162 + 178 = 1987 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152+ 158 + 162 + 178 + 192 = 2179 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152+ 158 + 162 + 178 + 192 + 198 = 2377 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152+ 158 + 162 + 178 + 192 + 198 + 204 + 206 + 208 + 214 =
3209 is prime
1+ 2+ 4+ 6+ 12+ 14+ 60+ 64+ 96+ 100+ 102+ 108+ 114+ 122+ 124+ 130+
132+ 146+ 152+ 158 + 162 + 178 + 192 + 198 + 204 + 206 + 208 + 214 +
216 + 296 + 308 + 326 + 328 + 330 + 332 + 334 + 342 + 350 + 356 + 358
+ 426 + 446 + 458 + 460 + 464 + 480 + 484 = 10303 is prime




There are squares in the sequence, 1, 25, 3721 = 61 ^ 2