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A096918 Intermediate prime factor of n-th product of 3 distinct primes.

%I A096918 #19 Nov 03 2024 02:03:48
%S A096918 3,3,3,5,3,3,5,5,3,5,3,7,5,5,3,7,3,5,5,3,5,7,7,3,5,3,7,7,3,5,11,5,5,3,
%T A096918 7,5,3,7,3,5,11,7,7,3,7,5,11,3,11,5,7,5,3,13,7,5,5,3,7,13,3,11,7,5,3,
%U A096918 5,11,7,3,5,7,13,7,3,7,5,5,3,11,11,3,5,17,7,3,7,13,7,5,3,11,5,5,11,5
%N A096918 Intermediate prime factor of n-th product of 3 distinct primes.
%H A096918 Amiram Eldar, <a href="/A096918/b096918.txt">Table of n, a(n) for n = 1..10000</a>
%F A096918 A096917(n)*a(n)*A096919(n) = A007304(n).
%F A096918 A096917(n) < a(n) < A096919(n).
%t A096918 f[n_]:=Last/@FactorInteger[n]=={1,1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];f3[n_]:=First/@FactorInteger[n][[2,1]];lst={};Do[If[f[n],AppendTo[lst,f3[n]]],{n,0,7!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 10 2010 *)
%o A096918 (Python)
%o A096918 from math import isqrt
%o A096918 from sympy import primepi, primerange, integer_nthroot, primefactors
%o A096918 def A096918(n):
%o A096918     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 A096918     def bisection(f,kmin=0,kmax=1):
%o A096918         while f(kmax) > kmax: kmax <<= 1
%o A096918         while kmax-kmin > 1:
%o A096918             kmid = kmax+kmin>>1
%o A096918             if f(kmid) <= kmid:
%o A096918                 kmax = kmid
%o A096918             else:
%o A096918                 kmin = kmid
%o A096918         return kmax
%o A096918     return sorted(primefactors(bisection(f)))[1] # _Chai Wah Wu_, Aug 30 2024
%Y A096918 Cf. A007304, A096917, A096919.
%K A096918 nonn
%O A096918 1,1
%A A096918 _Reinhard Zumkeller_, Jul 15 2004