Not So Fortunate Numbers

A Fortunate number, named after Reo Fortune, for a given positive integer n is the smallest integer m > 1 such that p_n# + m is a prime number, where the primorial p_n# is the product of the first n prime numbers.

For example, p₇# = 2×3×5×7×11×13 = 510,510. The smallest prime number after 510,511 is 510,529. Thus, 510,529 - 510,510 = 29 is a Fortunate number.

Fortune's conjecture states that no Fortunate number is composite. A Fortunate prime is a Fortunate number which is also a prime number. As of 2009, all the known Fortunate numbers are also Fortunate primes.

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Instead of only using primorial numbers in our sequence, what happens when we use all numbers such that for the largest prime dividing each number in our sequence, the primorial of that prime also divides that number (i.e., 630 = 2×3²×5×7 belongs to the sequence since p₄# = 210 divides 630)?

512 → 2⁵, 16,384 → 2⁹, and 524,288 → 2¹⁴ fail to generate Fortunate primes.

Do only perfect powers of two fail to generate Fortunate primes? Is there a pattern?

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For all numbers less than a billion in our sequence, the following fail to generate a Fortunate prime:

F#  Generator
9      512
9      8,388,608
15    4,194,304
15    67,108,864
21    524,288
25    147,456
25    373,248
25    393,216
25    1,062,882
25    1,259,712
25    4,251,528
25    4,718,592
25    5,308,416
25    5,971,968
25    10,077,696
25    17,915,904
25    21,233,664
25    35,831,808
25    42,467,328
25    172,186,884
27    16,384
35    1,119,744
35    1,492,992
35    33,554,432
35    47,775,744
35    150,994,944
35    362,797,056
49    22,118,400
49    31,104,000
49    32,805,000
49    42,187,500
49    56,623,104
49    90,000,000
49    286,654,464
49    364,500,000
49    720,000,000
49    859,963,392
55    679,477,248
65    10,616,832
77    159,252,480
77    188,956,800
85    322,486,272
119  314,572,800