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

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, 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, 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

Offset

1,1

Comments

A096917(n)*a(n)*A096919(n) = A007304(n); A096917(n) < a(n) < A096919(n).

Links

Table of n, a(n) for n=1..95.

Mathematica

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 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A096918(n):
    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)))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return sorted(primefactors(bisection(f)))[1] # Chai Wah Wu, Aug 30 2024

Crossrefs

Sequence in context: A162022 A318240 A262289 * A075018 A324974 A125958

Adjacent sequences: A096915 A096916 A096917 * A096919 A096920 A096921

Keyword

nonn

Author

Reinhard Zumkeller, Jul 15 2004

Status

approved