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%I #13 Aug 31 2024 01:57:29
%S 2,2,2,2,2,2,3,2,2,2,2,2,3,2,2,2,2,2,3,2,2,3,2,2,3,2,2,3,2,3,2,2,2,2,
%T 2,3,2,3,2,2,2,5,3,2,2,2,2,2,3,2,2,3,2,2,5,3,2,2,3,2,2,2,2,2,2,3,3,2,
%U 2,2,5,2,2,2,3,2,3,2,3,2,2,3,2,3,2,2,3,5,2,2,2,3,2,5,2,3,2,2,3,2,3,2
%N Smallest prime factor of n-th product of 3 distinct primes.
%C a(n)*A096918(n)*A096919(n) = A007304(n); a(n) < A096918(n) < A096919(n);
%C a(n) = A020639(A007304(n)).
%t f[n_]:=Last/@FactorInteger[n]=={1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f[n],AppendTo[lst,f1[n]]],{n,0,7!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 10 2010 *)
%o (Python)
%o from math import isqrt
%o from sympy import primepi, primerange, integer_nthroot, primefactors
%o def A096917(n):
%o def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
%o def bisection(f,kmin=0,kmax=1):
%o while f(kmax) > kmax: kmax <<= 1
%o while kmax-kmin > 1:
%o kmid = kmax+kmin>>1
%o if f(kmid) <= kmid:
%o kmax = kmid
%o else:
%o kmin = kmid
%o return kmax
%o return min(primefactors(bisection(f))) # _Chai Wah Wu_, Aug 30 2024
%K nonn,changed
%O 1,1
%A _Reinhard Zumkeller_, Jul 15 2004
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