
COMMENTS

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^611), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1<=n<=4, but composite for 5<=n<=17. The status of A077586(18) = 2^(2^611)1 is unknown. It is conjectured that A077586(n) is composite for all n>=5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^891).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568.  Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743.  Chai Wah Wu, Oct 16 2019
