The primorial \$p_n\#\$ is the product of the first \$n\$ primes. The sequence begins \2,ドル 6, 30, 210, 2310\$.

A Fortunate number, \$F_n\$, is the smallest integer \$m > 1\$ such that \$p_n\# + m\$ is prime. For example \$F_7 = 19\$ as:

$$p_7\# = 2\times3\times5\times7\times11\times13\times17 = 510510$$

Adding each number between \2ドル\$ and \18ドル\$ to \510510ドル\$ all yield composite numbers. However, \510510ドル + 19 = 510529\$ which is prime.

The Fortunate numbers below \200ドル\$ are

$3,ドル 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199$$

We'll say an integer \$n\$ is a "Fortunate sum" if it can be expressed as the sum of two distinct Fortunate numbers. For example, \22ドル = 3 + 19 = 5 + 17\$, so \22ドル\$ can be expressed as the sum of two Fortunate numbers, and so is a "Fortunate sum"

You are to take an integer \$n\$ as input and output a truthy value if \$n\$ is a Fortunate sum and a falsey value otherwise. You may swap the order (falsey indicates it is a Fortunate sum) if you wish. You may take input and output in any convenient format.

This is code-golf so the shortest code in bytes wins

Test cases

The first line is the Fortunate sums less than 100 (truthy values) and the second are the integers less than or equal to 100 that aren't Fortunate sums

8 10 12 16 18 20 22 24 26 28 30 32 36 40 42 44 50 52 54 56 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98
1 2 3 4 5 6 7 9 11 13 14 15 17 19 21 23 25 27 29 31 33 34 35 37 38 39 41 43 45 46 47 48 49 51 53 55 57 58 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 100