almost_primes.sf

#!/usr/bin/ruby

# a(n) is the smallest n-gonal number with exactly n prime factors (counted with multiplicity).
# https://oeis.org/A358863

# Known terms:
#   28, 16, 176, 4950, 8910, 1408, 346500, 277992, 7542080, 326656, 544320, 120400000, 145213440, 48549888, 4733575168, 536813568, 2149576704, 3057500160, 938539560960, 1358951178240

# PARI/GP program:
#`(

bigomega_polygonals(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(ispolygonal(t,k), listput(list, t))), forprime(q = p, sqrtnint(B\m, n), my(t=m*q); if(ceil(A/t) <= B\t, list=concat(list, f(t, q, n-1))))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(k < 3, return()); my(x=2^n, y=2*x); while(1, my(v=bigomega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~

)

func upper_bound(n, from = 2, upto = 2*from) {

    say "\n:: Searching an upper-bound for a(#{n})\n"

    loop {

        var count = n.almost_prime_count(from, upto)

        if (count > 0) {

            say "Sieving range: [#{from}, #{upto}]"
            say "This range contains: #{count.commify} elements\n"

            n.almost_primes_each(from, upto, {|v|
                if (v.is_polygonal(n)) {
                    say "a(#{n}) = #{v}"
                    return v
                }
            })
        }

        from = upto+1
        upto *= 2
    }
}

upper_bound(29)

__END__
a(3) = 28
a(4) = 16
a(5) = 176
a(6) = 4950
a(7) = 8910
a(8) = 1408
a(9) = 346500
a(10) = 277992
a(11) = 7542080
a(12) = 326656
a(13) = 544320
a(14) = 120400000
a(15) = 145213440
a(16) = 48549888
a(17) = 4733575168
a(18) = 536813568
a(19) = 2149576704
a(20) = 3057500160
a(21) = 938539560960
a(22) = 1358951178240
a(23) = 36324805836800
a(24) = 99956555776
a(25) = 49212503949312
a(26) = 118747221196800
a(27) = 59461613912064
a(28) = 13749193801728
a(29) = 7526849672380416
a(30) = 98516240758210560
a(31) = 4969489493917696
a(32) = 78673429816934400
a(33) = 4467570822566903808
a(34) = 1013309912383488000