experimental-projects/oeis-research/Daniel Suteu/Smallest number k > 1 such that bigomega(k^n - 1) = n/upper-bounds.sf

#!/usr/bin/ruby

# a(n) is the smallest number k > 1 such that bigomega(k^n - 1) = n.
# https://oeis.org/A368163

# Known terms:
#   3, 4, 4, 10, 17, 8, 25, 5, 28, 9, 81, 13, 289, 64, 100, 41, 6561, 31, 6657, 57, 529, 1025

# a(24) = 79; a(27) = 961; a(28) = 729; a(30) = 361; a(32) = 2047.

# Lower-bounds:
#   a(23) > 40448
#   a(25) > 22720
#   a(26) > 7768

# Conjectured lower-bounds:
#   a(23) > 57344
#   a(25) > 24576

# Upper-bounds:
#   a(23) <= 286721 (found by Jon E. Schoenfield, Sep 25 2018)
#   a(23) <= 196609
#   a(25) <= 28561
#   a(26) <= 14015

# Close call:
#   bigomega(98305^23 - 1) = 22

# Strong-candidates for a(23) (not yet factorized):
#   131073,

# Cf. A241793.

Num!VERBOSE=true
Num!USE_FACTORDB=true
Num!USE_CONJECTURES=true

func a(n, from=1) {
    for k in (from..Inf) {

        #k.is_prime && next
        #is_prob_squarefree(k**n - 1, 1e3) && next

        say "[#{n}] Testing: #{k}"

        try {
            Sig.ALRM {
                die "alarm\n"
            }
            Sys.alarm(60)
            if (is_almost_prime(k**n - 1, n)) {
                say "a(#{n}) = #{k}"
                return k
            }
            Sys.alarm(0)
        }
        catch {
            say "Timeout..."
            Sys.run("killall yafu")
        }
    }
}

say a(26)