A180065 - OEIS

%I #13 Mar 05 2023 01:37:14

%S 2047,15841,800605,293609485,10761055201,5478598723585,

%T 713808066913201,90614118359482705,5993318051893040401,

%U 24325630440506854886701,27146803388402594456683201,4365221464536367089854499301,2162223198751674481689868383601,548097717006566233800428685318301

%N Smallest strong pseudoprime to base 2 with n prime factors.

%H Daniel Suteu, <a href="/A180065/b180065.txt">Table of n, a(n) for n = 2..17</a>

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%e 800605 is the third term because 800605 = 5 * 13 * 109 * 113, more prime factors than smaller 2-strong pseudoprimes.

%o (PARI)

%o strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;

%o strong_fermat_psp(A, B, k, base) = A=max(A, vecprod(primes(k))); (f(m, l, lo, k, e, r) = 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 && strong_check(p, base, e, r), 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; strong_check(p, base, e, r) || 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, e, r)); q *= p; Mod(base, q)^z == 1 || break; v *= p))); list); my(res=f(1, 1, 2, k, 0, 1)); for(v=0, logint(B, 2), res=concat(res, f(1, 1, 2, k, v, -1))); vecsort(Set(res));

%o a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, n, 2)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Mar 04 2023

%Y Cf. A007011, A001262.

%K nonn

%O 2,1

%A Kevin Batista (kevin762401(AT)yahoo.com), Aug 09 2010

%E a(6)-a(10) and editing from _Charles R Greathouse IV_, Aug 29 2010

%E a(11)-a(15) from _Daniel Suteu_, Sep 24 2022