The number 37778715690312487141376 is also in the sequence.  Daniel Suteu, Jan 27 2020
The first 3 terms have the form (2^p1)*(2^(p1))*((2^p1)^22), i.e., a Perfect number times a Carol prime.  G. L. Honaker, Jr., Jan 27 2020
In other words, the values of p are given by the intersection of A091515 and A000043. Currently, only four such values of p are known: {2, 3, 7, 19}.  Daniel Suteu, Jan 27 2020
From Bernard Schott, Jan 28 2020: (Start)
Proposition: If a number N_p is of the form Q_p * C_p where Q_p = (2^(p1)) * (2^p  1) is a perfect number and C_p = (2^p1)^22 is a Carol prime then, the sum of the nonprime proper divisors of N_p called S_p(N_p) is equal to N_p.
Proof:
The sum of the nonprime proper divisors of N_p is:
S_p(N_p) = (2* Q_p  2  (2^p1)) + ((Q_p  1) * C_p).
In the first parenthesis, there is the sum of the nonprime proper divisors of N_p coming only from the perfect number Q_p, then in the second parenthesis, there is the sum of the nonprime proper divisors of N_p coming from C_p.
Then, this sum of the nonprime proper divisors of N_p, S_p(N_p) is indeed equal to N_p = (2^(p1)) * (2^p1) * ((2^p1)^22).
Hence, (2^191)*(2^(191))*((2^191)^22) = 37778715690312487141376 is a term. (End)
10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957.  Giovanni Resta, Jan 28 2020
