prog.sf

#!/usr/bin/ruby

# Smallest odd primitive abundant number (A006038) having n distinct prime factors.
# https://oeis.org/A188342

# Known terms:
#   945, 3465, 15015, 692835, 22309287, 1542773001, 33426748355, 1635754104985, 114761064312895, 9316511857401385, 879315530560980695

# Upper-bounds due to Donovan Johnson for n=14 (Mar 31 2011) and M. F. Hasler for n=15..17 (Jul 17 2016)
#   a(14) <= 88452776289145528645
#   a(15) <= 2792580508557308832935
#   a(16) <= 428525983200229616718445
#   a(17) <= 42163230434005200984080045

# Extra upper-bounds (generated with this program):
#   a(18) <= 1357656019974967471687377449
#   a(19) <= 189407457935656632167109232619
#   a(20) <= 25557786985317796527581223227267
#   a(21) <= 3791929072764979008305867397500527
#   a(22) <= 456592871291171295882477091922563457

# For n = 18 and lpf(N) = 5, we also have:
#   a(18) <= 4506505276387497069886673045

var min = Inf
var max = 456592871291171295882477091922563457

var omega_value = 22        # exact number of prime factors

var lpf = 7                 # exactly smallest prime factor of n
var gpf_max = 103           # limit for the largest prime of n
var max_pi_p_diff = 3       # limit for the maximum difference between primes (pi(gpf(n)) - pi(q))

var min_abundancy = 2       # minimum abundancy
var max_abundancy = 2.1     # maximum abundancy

func is_primitive_abundant(n) {
    n.factor.none {|p|
        abundancy(n/p) > 2
    }
}

func f(n, p) {

    if (n.omega > omega_value) {
        return nil
    }

    var ab = n.abundancy

    if (ab > max_abundancy) {
        return nil
    }

    if (ab >= min_abundancy) {
        if (n<min && n.omega==omega_value && is_primitive_abundant(n)) {
            min = n
            say [p, omega(n), n]
        }
    }

    var q = p.next_prime

    if (q > gpf_max) {
        return nil
    }

    if (q.pi - p.pi <= max_pi_p_diff) {
        if (n*q <= max) {
            f(n*q, q)
        }
    }

    f(n, q)
}

say f(lpf, lpf)