[seqfan] Primes that are the sum of 4 distinct primes; and Primes that are not the sum of 4 distinct primes

[Jonathan Post]

Substitute "natural number" for "prime" and we get a seq that njas
rightly calls too (initially) similar to the natural numbers.

Is this subset interesting?

Primes that are the sum of 4 distinct primes (in at least one way, is
it interesting to enumerate how many ways?):

17, 23, 29, 31, 37, 37, 41, 43, 47, 53, 59

Examples:

17 = 2+3+5+7
23 = 2+3+5+13 = 2+3+7+11
29 = 2+3+5+19 = 2+3+7+17 = 2+3+11+13
31 = 2+3+7+19 = 2+5+7+17 = 2+5+11+13
37 = 2+3+13+19 = 2+5+7+23 = 2+5+11+19 = 2+5+13+17 = 2+7+11+17
41 = 2+3+5+31 = 2+3+7+29 = 2+3+13+23 = 2+5+11+23 = 2+7+13+19
43 = 2+3+7+31 = 2+5+7+29 = 2+5+13+23 = 2+5+17+19 = 2+7+11+23 = 2+11+13 + 17
47 = 2+3+5+37 = 2+3+11+31 = 2+3+13+29 = 2+5+11+29 = 2+5+17+23
53 = 2+3+5+43 = 2+3+7+41 = 2+3+11+37 = 2+5+17+29 = 2+7+13+31 = 2+11+17 + 23
59 = 2+3+7+47 = 2+3+11+43 = 2+3+13+41 = 2+5+11+41 = 2+5+23+29 =
2+7+13+37 = 2+7+19+31 = 2+11+17 + 29
61 = 2+3+13+43 = 2+5+7+47 = 2+5+11+43 = 2+5+13+41 = 2+5+17+37 =
2+5+23+31 = 2+7+11+41 = 2+7+23+29 = 2+11+17 + 31 = 2+11+19 + 29 =
2+13+17 + 29 = 2+17+19+23
67 = 2+5+7+53 = 2+5+13+47 = 2+5+17+43 = 2+5+19+41 = 2+5+23+37 =
2+7+11+47 = 2+7+17+41 = 2+11+13 + 41 = 2+11+17 + 37 = 2+13+23 + 29 =
2+17+19+29
71 = 2+3+5+61 = 2+3+7+59 = 2+5+11+53 = 2+5+17+47 = 2+5+23+41 =
2+7+19+43 = 2+11+17 + 41 = 2+13+19 + 37
73 = 2+5+7+59 = 2+5+13+53 = 2+7+11+53 = 2+7+23+41 = 2+11+17 + 43 =
2+11+19 + 41 = 2+13+17 + 41 = 2+19+23 + 29
79 = 2+5+19+53 = 2+5+23+43 = 2+7+11+59 = 2+7+17+53 = 2+7+23+47 =
2+11+13 + 53 = 2+13+17 + 47 = 2+13+23 + 41 = 2+17+19+41

and so on (above is incomplete).

Obviously, the sum of 4 odd primes is even, and thus composite, hence
2 must be one of these 4 distinct primes in the sums.

The complement (as a subset of primes):

Primes that are not the sum of 4 distinct primes
2, 3, 5, 7, 11, 13, 19 (what's the next value, if any?)

It is premature for me to submit these, as I have barely spent an hour
yet on them, and thus I ask for the opinion of those who have not
retired from programming.

Thank you for your attention and consideration,

Jonathan Vos Post