A096917 - OEIS

A096917

Smallest prime factor of the n-th product of 3 distinct primes.

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, 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, 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

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OFFSET

1,1

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

a(n)*A096918(n)*A096919(n) = A007304(n).

a(n) < A096918(n) < A096919(n).

a(n) = A020639(A007304(n)).

MATHEMATICA

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 *)

PROG

(Python)

from math import isqrt

from sympy import primepi, primerange, integer_nthroot, primefactors

def A096917(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 min(primefactors(bisection(f))) # Chai Wah Wu, Aug 30 2024

CROSSREFS

Cf. A020639, A007304, A096918, A096919.