prog.sf

#!/usr/bin/ruby

# a(n) = smallest pseudoprime to base 2 with n prime factors.
# https://oeis.org/A007011

# Known terms:
#   341, 561, 11305, 825265, 45593065, 370851481, 38504389105, 7550611589521, 277960972890601, 32918038719446881, 1730865304568301265, 606395069520916762801, 59989606772480422038001, 6149883077429715389052001, 540513705778955131306570201, 35237869211718889547310642241

# New terms found:
#   a(18) = 13259400431578770557375638157521
#   a(19) = 580827911915963785382253469211401
#   a(20) = 292408776547176479576081475390697801
#   a(21) = 39498823114155235923831808284152901601
#   a(22) = 3284710806953570725820888298298841565601
#   a(23) = 327373784481535488655521620744179013043601
#   a(24) = 221404014114397213806721960178887462402717201
#   a(25) = 43691666165877828056799483424028795272585383601
#   a(26) = 13213974925373194568934435211730355813060799098001
#   a(27) = 1952204134080476076724242017017925744953021675628161

#`(

# 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)));
a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=fermat_psp(x, y, n, 2)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~


# 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)));
a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=fermat_psp(x, y, n, 2)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ ~~~~

)

func T(k, base=2) {
    var x = pn_primorial(k)
    var y = 3*x

    loop {
        var arr = k.fermat_psp(base, x, y)
        if (arr) {
            return arr[0]
        }

        x = y+1
        y = 3*x
    }
}

for n in (2..15) {
    say "a(#{n}) = #{T(n)}"
}