prog.sf

#!/usr/bin/ruby

# a(n) is the least number such that the concatenation of all the previous terms with it is squarefree and has n prime factors. a(0)=0.
# https://oeis.org/A309234

# Known terms:
#   0, 2, 1, 10, 29, 5, 154, 66, 1005, 158, 18634, 8190

# Incorrect terms:
#   285510, 196790, 847630

# Corrected terms:
#   a(12) = 113022
#   a(13) = 62010
#   a(14) = 102310
#   a(15) = 5313758
#   a(16) = 15617985
#   a(17) = 510510

# Lower-bounds:
#   a(18) > 51844464

# PARI/GP program:
#   a(n) = if(n==0, return(0)); my(prefix=vector(n-1, k, Str(a(k)))); for(k=1, oo, my(t=eval(concat(concat(prefix, [Str(k)])))); if(issquarefree(t) && bigomega(t) == n, return(k))); \\ ~~~~

# PARI/GP program (v2):
#   my(n=1,L=[]); print1("0, "); while(1, for(k=1, oo, my(t=eval(concat(concat(L, [Str(k)])))); if(issquarefree(t) && bigomega(t) == n, print1(k, ", "); L=concat(L, [Str(k)]); n += 1; break))); \\ ~~~~

func prefix(n) {

    var known = [0, 2, 1, 10, 29, 5, 154, 66, 1005, 158, 18634, 8190, 113022, 62010, 102310, 5313758, 15617985, 510510]

    if (n <= known.end) {
        return known[n]
    }

    die "Too large: n = #{n}"
}

func a(n, from=0) {

    return 0 if (n == 0)

    var v = (^n -> map(prefix).join)

    for k in (from..Inf) {
        say "Checking: #{k}"
        if (Num(v + k).is_squarefree_almost_prime(n)) {
            say "Found: a(#{n}) = #{k}"
            return k
        }
    }
}

say a(18, 51844464)