possible seq: Partial sum of A061712

[Jonathan Post]

Partial sum of A061712  Smallest prime with Hamming weight n (i.e.
with exactly n 1's when written in binary).

2, 5, 12, 35, 66, 377, 504, 887, 1878, 3917, 7500, 13643, 21834,
95561, 159048, 288071, 419142, 941381, 1465668, 3431747, 7560514,
23811393, 38491456, 72037695, 139146046, 340210493, 600257340,
1136997179, 2210476858, 7545631545, 9693115192, 18274661175,
35186094902, 67398349621, 118937957172, 325079610163, 591367582514,
1622158684977, 2446792405808, 4643668177711

I'm interested in which values of the partial sum (beyond the first
two, a(8), a(10), and a(14)) are themselves prime. The sequence,
obviously,  is strictly increasing, and alternates between even and
odd.

Example:

a(40) =
2 + 3 + 7 + 23 + 31 + 311 + 127 + 383 + 991 + 2039 + 3583 + 6143 +
8191 + 73727 + 63487+ 129023 + 131071+ 522239 + 524287+ 1966079 +
4128767 + 16250879 + 14680063 + 33546239 + 67108351 + 201064447 +
260046847 + 536739839 + 1073479679 + 5335154687 + 2147483647 +
8581545983 + 16911433727 + 32212254719 + 51539607551 + 206141652991 +
266287972351 + 1030791102463 + 824633720831 + 2196875771903 =
4643668177711 is prime.