experimental-projects/oeis-research/Daniel Suteu/Smallest overpseudoprime to base 2 (A141232) with n distinct prime factors/upper-bounds.sf

#!/usr/bin/ruby

# Smallest overpseudoprime to base 2 (A141232) with n distinct prime factors.
# https://oeis.org/A353409

# Known terms:
#   2047, 13421773, 14073748835533

# Upper-bounds:
#   a(5) <= 1376414970248942474729
#   a(6) <= 48663264978548104646392577273
#   a(7) <= 294413417279041274238472403168164964689
#   a(8) <= 98117433931341406381352476618801951316878459720486433149
#   a(9) <= 1252977736815195675988249271013258909221812482895905512953752551821

include("../../../factordb/auto.sf")

var min = Inf

#~ var n = 5
#~ var psize = 9
#~ var from = 1206

var n = 6
var psize = 9
var from = 879

#~ var n = 7
#~ var psize = 10
#~ var from = 620
#~ #var from = 1200

#~ var n = 8
#~ var psize = 11
#~ var from = 369636

say ":: Searching upper-bounds for n = #{n} from k = #{from}"

var counter = 0

for k in (from .. from+1e3) {

    if (++counter % 10 == 0) {
        say ":: Checking: k = #{k}"
    }

    # Conjecture: the ord(2, a(n)) must be of this form
    k.is_prime || is_prime(k/4) || is_prime(k/12) || next

    var f = factordb(2**k - 1).grep{ .len <= psize }.grep{.is_prime}.grep { powmod(2, k, _) == 1 }.grep{ znorder(2,_) == k }

    say "[#{k}] Binomial: #{binomial(f.len, n)}" if (f.len > n)

    var count = 0

    f.combinations(n, {|*a|
        var t = a.prod

        if (t.is_strong_psp) {
            if (t < min) {
                say "a(#{n}) <= #{t}"
                min = t
            }
        }

        break if (++count > 1e4)
    })
}

__END__
a(5) <= 3223802185639011132549803
a(5) <= 636607858967934928371769
a(5) <= 124250696089090697678753
a(5) <= 8278905362357819790631
a(5) <= 1376414970248942474729

a(6) <= 32245825439777493648426550929515449
a(6) <= 721606983841657320586259138751241
a(6) <= 48663264978548104646392577273

a(7) <= 294413417279041274238472403168164964689
a(8) <= 98117433931341406381352476618801951316878459720486433149
a(9) <= 1252977736815195675988249271013258909221812482895905512953752551821