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A361256 Smallest base-n strong Fermat pseudoprime with n distinct prime factors.

%I A361256 #7 Mar 26 2023 16:53:42
%S A361256 2047,8911,129921,381347461,333515107081,37388680793101,
%T A361256 713808066913201,665242007427361,179042026797485691841,
%U A361256 8915864307267517099501,331537694571170093744101,2359851544225139066759651401,17890806687914532842449765082011
%N A361256 Smallest base-n strong Fermat pseudoprime with n distinct prime factors.
%C A361256 Main diagonal of A360184.
%o A361256 (PARI)
%o A361256 strong_check(p, base, e, r) = my(tv=valuation(p-1, 2)); tv > e && Mod(base, p)^((p-1)>>(tv-e)) == r;
%o A361256 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 A361256 a(n) = if(n < 2, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=strong_fermat_psp(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x);
%Y A361256 Cf. A001262, A180065, A271874, A360184.
%K A361256 nonn
%O A361256 2,1
%A A361256 _Daniel Suteu_, Mar 06 2023