A023567

[Jack Brennen]

This PARI/GP fragment matches the first twenty members of the sequence
as shown...

  vector(20,x,c=0;j=x;t=3;while(j>1,c+=prime(j-1);j-=t;t+=1);c)

Basically, the general form of a member of the sequence seems to be:
a prime plus the third previous prime, plus the fourth previous
prime before that, plus the fifth previous prime before that, etc.

So 176 == 67+53+37+17+2.

I didn't get that at all from the description, just figured out on
my own how to duplicate the shown sequence.  And of course, I could
be wrong and just found a sequence which is coincidentally similar.