sidef-scripts/Math/fermat_pseudoprimes_in_range.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 30 August 2022
# https://github.com/trizen

# Generate all the k-omega Fermat pseudoprimes in range [a,b]. (not in sorted order)

# Definition:
#   k-omega primes are numbers n such that omega(n) = k.

# See also:
#   https://en.wikipedia.org/wiki/Almost_prime
#   https://en.wikipedia.org/wiki/Prime_omega_function
#   https://trizenx.blogspot.com/2020/08/pseudoprimes-construction-methods-and.html

# PARI/GP program (slow):
#   fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(base%q != 0, my(v=m*q, t=q, r=nextprime(q+1)); while(v <= B, my(L=lcm(l, znorder(Mod(base, t)))); if(gcd(L, v) == 1, if(j==1, if(v>=A && if(k==1, !isprime(v), 1) && (v-1)%L == 0, listput(list, v)), if(v*r <= B, list=concat(list, f(v, L, r, j-1)))), break); v *= q; t *= q))); list); vecsort(Vec(f(1, 1, 2, k)));

# PARI/GP program (fast):
#   fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k) = my(list=List()); my(hi=sqrtnint(B\m, k)); if(lo > hi, return(list)); if(k==1, forstep(p=lift(1/Mod(m, l)), hi, l, if(isprimepower(p) && gcd(m*base, p) == 1, my(n=m*p); if(n >= A && (n-1) % znorder(Mod(base, p)) == 0, listput(list, n)))), forprime(p=lo, hi, base%p == 0 && next; my(z=znorder(Mod(base, p))); gcd(m,z) == 1 || next; my(q=p, v=m*p); while(v <= B, list=concat(list, f(v, lcm(l, z), p+1, k-1)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); vecsort(Set(f(1, 1, 2, k)));

func fermat_pseudoprimes_in_range(A, B, k, base, callback) {

    A = max(k.pn_primorial, A)

    var seen = Hash()

    func (m, L, lo, j) {

        var hi = idiv(B,m).iroot(j)

        lo > hi && return nil

        if (j == 1) {

            if (L == 1) {   # optimization
                each_prime(lo, hi, {|p|

                    p.divides(base) && next

                    for (var v = (m == 1 ? p*p : m*p); v <= B; v *= p) {
                        v >= A         || next
                        v.is_psp(base) || break
                        callback(v) if !(seen{v} := 0 ++)
                    }
                })
                return nil
            }

            var t = m.invmod(L)

            t > hi && return nil
            t += L*idiv_ceil(lo - t, L) if (t < lo)
            t > hi && return nil

            for p in (range(t, hi, L)) {

                p.is_prime_power   || next
                p.is_coprime(m)    || next
                p.is_coprime(base) || next

                var v = m*p
                v >= A || next

                if (znorder(base, p) `divides` v.dec) {
                    callback(v) if !(seen{v} := 0 ++)
                }
            }

            return nil
        }

        each_prime(lo, hi, {|p|

            p.divides(base) && next

            var z = znorder(base, p)
            m.is_coprime(z) || next

            var v = m*p
            var q = p

            while (v <= B) {
                __FUNC__(v, lcm(L, z), p+1, j-1)
                q *= p
                powmod(base, z, q) == 1 || break
                v *= p
            }
        })
    }(1, 1, 2, k)

    return callback
}

# Generate all the Fermat pseudoprimes to base 3 in range [1, 10^4]

var from = 1
var upto = 1e4
var base = 3

var arr = gather {
    for k in (1..100) {
        break if (k.pn_primorial > upto)
        fermat_pseudoprimes_in_range(from, upto, k, base, { take(_) })
    }
}

say arr.sort

# Run some tests

if (false) {    # true to run some tests
    for k in (1..5) {

        say "Testing: k = #{k}"

        var lo = k.pn_primorial
        var hi = lo*1000

        for base in (2..100) {

            var this = k.fermat_psp(base, lo, hi)
            #var this = k.omega_primes(lo,hi).grep{.is_psp(base) && .is_composite }
            var that = gather {
                fermat_pseudoprimes_in_range(lo, hi, k, base, func (n) { take(n) })
            }.sort

            this == that ||
                die "Error for k = #{k} and base = #{base} with hi = #{hi}\n#{this} != #{that}";
        }
    }
}

__END__
[91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911]