This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For exponent sum n = 3 no prime exists, (3*5*7-1)/2 = 52 is composite. a(4) = 157: (3^2*5*7-1)/2 is the least prime with exponent sum n = 4. a(5) = 787: there are 6 ways to choose the exponents of 3, 5, 7 with sum n = 5, i.e., [3,1,1], [1,3,1], [1,1,3], [2,2,1], [2,1,2], [1,2,2]. (3^3*5*7-1)/2 = 472 is composite, but (3^2*5^2*7-1)/2 = 787 is prime.
Table[Min[Select[(3^#[[1]] 5^#[[2]] 7^#[[3]]-1)/2&/@Flatten[Permutations/@ IntegerPartitions[n,{3}],1],PrimeQ]],{n,30}]/.\[Infinity]->Nothing (* Harvey P. Dale, Oct 29 2020 *)
seqpp(np0,np,add,lim) = {for(m=np0,lim, my(pmin=oo); forpart(V=m, forperm(np,P, my(p=(prod(k=1,np,prime(k+1)^V[P[k]])+add)/2); if(isprime(p), pmin=min(pmin,p))), [1,m-np+1],[np,np]); print1(pmin,", "))};
seqpp(4,3,-1,27)
a(3) = 53: (3*5*7+1)/2 = 106/2 is prime. a(4) = 263: The first choice of exponents leads to the composite (3^2*5*7+1)/2 = 158, but the next choice (3*5^2*7+1)/2 = 526/2 is prime.
seqpp (3,3,1,27) \\ using function seqpp defined in A337427