omega_primes.sf

#!/usr/bin/ruby

# a(n) is the smallest centered n-gonal number with exactly n distinct prime factors.
# https://oeis.org/A358894

# Known terms:
#   460, 99905, 463326, 808208947, 23089262218

# PARI/GP program:
#`(

# Version 1
omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(s=q%k); if(s==1 || s==7 || s==9 || s==15, my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A, if (issquare((8*(v-1))/k + 1) && ((sqrtint((8*(v-1))/k + 1)-1)%2 == 0), listput(list, v))), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~

# Version 2
omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(s=q%k); if(s==1 || s==9, my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A, if (issquare((8*(v-1))/k + 1) && ((sqrtint((8*(v-1))/k + 1)-1)%2 == 0), listput(list, v))), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~

)

# Find the residues of the prime factors of centered n-gonal numbers:
#   for n in (3..20) { say [n, 1000.of {|k| ((n*k*(k+1))/2 + 1) }.map{.factor.map{.mod(n)}}.flat.uniq.sort] }

# Resides for each n = 3..n:
#`(
    [3, [1, 2]]
    [4, [1]]
    [5, [1, 2, 3, 4]]
    [6, [1]]
    [7, [1, 2, 4]]
    [8, [1, 3, 5, 7]]
    [9, [1, 2, 4, 5, 7, 8]]
    [10, [1, 9]]
    [11, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]]
    [12, [1, 11]]
    [13, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]
    [14, [1, 3, 5, 9, 11, 13]]
    [15, [1, 2, 4, 7, 8, 11, 13, 14]]
    [16, [1, 7, 9, 15]]
    [17, [1, 2, 3, 4, 8, 9, 13, 15, 16]]
    [18, [1, 5, 7, 11, 13, 17]]
    [19, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]]
    [20, [1, 3, 7, 9, 11, 13, 17, 19]]
)

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

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

    loop {

        var count = n.squarefree_almost_prime_count(from, upto)

        if (count > 0) {

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

            n.squarefree_almost_primes_each(from, upto, {|v|
                if ((n `divides` 8*v.dec) && (8*v.dec / n + 1 -> is_square)) {
                    say "a(#{n}) = #{v}"
                    return v
                }
            })
        }

        from = upto+1
        upto *= 2
    }
}

upper_bound(13)

__END__
a(3) = 460
a(4) = 99905
a(5) = 463326
a(6) = 808208947
a(7) = 23089262218
a(8) = 12442607161209225
a(9) = 53780356630
a(10) = 700326051644920151
a(11) = 46634399568693102
a(12) = 45573558879962759570353