experimental-projects/oeis-research/Michel Marcus/Least number k such that the concatenation of k composites from composite(n) is prime/prog.sf

#!/usr/bin/ruby

#~ f(n) = least number k such that the concatenation of k composite numbers starting from the n-th composite is prime

#~ f(1)  = ?
#~ f(2)  = ?
#~ f(3)  = 2      (3rd composite = 8 ; 89 is prime)
#~ f(4)  = ?
#~ f(5)  = 646
#~ f(6)  = 14662  (found by Hans Havermann via SeqFan)
#~ f(7)  = 6      (7th composite = 14 ; 141516182021 is prime)
#~ f(8)  = 14     (8th composite = 15 ; 1516182021222425262728303233 is prime)
#~ f(9)  = 302
#~ f(10) = 24     (10th composite = 18; 182021222425262728303233343536383940424445464849 is prime)

# Extra values:
#   f(11) = 1388  (found by cwwuieee via SeqFan)
#   f(12) = 6
#   f(13) = ?
#   f(14) = ?

# Bounds:
#   f(1) > composite_count(11333)
#   f(2) > composite_count(6230)
#   f(4) > composite_count(5212)

func isok(n, k) {
    ncomposites(k, n.composite).join.to_i.is_prob_prime
}

assert(isok(3, 2))
assert(isok(5, 646))
assert(isok(7, 6))
assert(isok(8, 14))
assert(isok(9, 302))
assert(isok(10, 24))
assert(isok(12, 6))

func f(n, i=n.composite) {

    var c   = n.composite
    var str = (c..i -> grep { .is_composite }.join)

    c = i

    loop {

        say "[n = #{n}] Testing: #{c} (len: #{str.len})"

        if (str.to_i.is_prob_prime) {
            var r = composite_count(n.composite, c)
            say "Found: f(#{n}) = #{r}"
            return r
        }

        c.next_composite!
        str += c.to_s
    }
}

#say f(16)
say f(1, 9938)

#say f(1, 6000)
#say f(5, 776)