S numbers and my quest for mathematical immortality

[SLD]

OK I was reading up a bit on Taxicab numbers which were created by Srinavasa Ramanujan. They should be called Ramanujan numbers, IMHO. But I was trying to think of other types of unique numbers for no good reason other than my quest for mathematical immortality- which is probably unlikely given that I am nowhere near the level of Ramanujan.

Nevertheless I decided to try and see what happens if you take primes, cube them and add them together. The first such number would be 16, ignoring 1 as a prime. That set doesn’t seem particularly interesting. Then I thought what if we always had to have three such numbers, the first then being 16. Still not interesting. Well, I could be wrong.

Then I decided that they must be three consecutive primes so the first would be 2 cubed plus 3 cubed plus 5 cubed, or 160. I kept going:

160
495
1799
3871
8441

Then I thought I wonder if any of these are primes. Nope. None are. Is that true for all? I plowed onwards:

13969
23939
43415
66347

Finally a prime number!

Therefore I have decided that henceforth, any number that is the sum of three consecutive primes that are each cubed that is also a prime number is a S number.

What’s the next S number?

 

[Swammerdami]

There appear to be 32 S numbers less than a billion.
Among these 903914731 (661³ + 673³ + 677³) is the largest and 199081 (37³ + 41³ + 43³) the 2nd smallest.