Primes suitable for nested exponentials

In general, if we want to handle several levels of nested exponentials, we will have to precompute several primes n₁, n₂, ... so that n_i+1 = 4k_in_i+1. This will make it possible to represent i at all levels. The values of k_i should be chosen as small as possible, as we want to keep the ratio between the largest and smallest primes small. The following Maple code generates a sequence of 5 primes with this property. It should be noticed that the ratio between n₅/n₁ = 1536. The minimum ratio, for all k_i=1, or 256 is not attainable.

n1 := 10^9:
maxn5 := n1*10^10:

to 10000 do
  n1 := nextprime(n1);
  if modp(n1,4) = 1 then for n2 from 4*n1+1 by 4*n1 while 64*n2+21 < maxn5 do
    if isprime(n2) then for n3 from 4*n2+1 by 4*n2 while 16*n3+5 < maxn5 do
      if isprime(n3) then for n4 from 4*n3+1 by 4*n3 while 4*n4+1 < maxn5 do
        if isprime(n4) then for n5 from 4*n4+1 by 4*n4 while n5 < maxn5 do
          if isprime(n5) then
            maxn5 := n5;
            lprint(n1,n2,n3,n4,n5)
          fi od fi od fi od fi od fi od:

The best set of primes found with the above is 1000077157 4000308629 48003703549 192014814197 1536118513577.