almost_primes.sf

#!/usr/bin/ruby

# a(n) is the smallest centered n-gonal number with exactly n prime factors (counted with multiplicity).
# https://oeis.org/A358926

# Known terms:
#   316, 1625, 456, 3964051, 21568, 6561, 346528

# PARI/GP program:
#    a(n) = if(n<3, return()); for(k=1, oo, my(t=((n*k*(k+1))/2+1)); if(bigomega(t) == n, return(t))); \\ ~~~~

# PARI/GP generation 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(issquare((8*(t-1))/k + 1) && ((sqrtint((8*(t-1))/k + 1)-1)%2 == 0), 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); \\ ~~~~

# Version with primes in certain residue classes
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(s=q%k); if(s==1 || s==7 || s==9 || s==15, my(t=m*q); if(issquare((8*(t-1))/k + 1) && ((sqrtint((8*(t-1))/k + 1)-1)%2 == 0), listput(list, t)))), forprime(q = p, sqrtnint(B\m, n), my(s=q%k); if(s==1 || s==7 || s==9 || s==15, 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_centered_polygonal(n)) {
                    say "a(#{n}) = #{v}"
                    return v
                }
            })
        }

        from = upto+1
        upto *= 2
    }
}

upper_bound(16)

__END__
a(3) = 316
a(4) = 1625
a(5) = 456
a(6) = 3964051
a(7) = 21568
a(8) = 6561
a(9) = 346528
a(10) = 3588955448828761
a(11) = 1684992
a(12) = 210804461608463437
a(13) = 36865024
a(14) = 835904150390625
a(15) = 2052407296