A084126 Prime factor <= other prime factor of n-th semiprime.

2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 2, 5, 2, 3, 2, 7, 3, 5, 3, 2, 2, 5, 3, 2, 7, 2, 5, 2, 3, 7, 3, 2, 5, 2, 3, 5, 2, 7, 11, 2, 3, 3, 7, 2, 3, 2, 11, 5, 2, 5, 2, 3, 7, 2, 13, 3, 2, 3, 5, 11, 2, 3, 2, 7, 5, 2, 11, 3, 2, 5, 7, 2, 3, 13, 2, 5, 3, 13, 3, 11, 2, 7, 2, 5, 3, 2, 2, 7, 17, 3, 5, 2, 13, 7, 2, 3, 5, 3, 2

Offset: 1

Author

Reinhard Zumkeller, May 15 2003

Keywords

• nonn

Comments

Lesser of the prime factors of A001358(n). - Jianing Song, Aug 05 2022

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A001358 (the semiprimes), A084127 (greater of the prime factors of the semiprimes).

Cf. A068318, A087718, A087794, A089994, A089995, A096916, A106550, A106554, A108541, A131284, A138510, A138511.

Programs

Haskell

a084126 = a020639 . a001358  -- Reinhard Zumkeller, Nov 25 2012

Mathematica

FactorInteger[#][[1,1]]&/@Select[Range[500],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 25 2018 *)

Python

from sympy import primepi, primerange, primefactors
def A084126(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    return min(primefactors(bisection(f,n,n))) # Chai Wah Wu, Apr 03 2025

Formula

a(n) = A020639(A001358(n)).

a(n) = A001358(n)/A006530(A001358(n)). [corrected by Michel Marcus, Jul 18 2020]

a(n) = A001358(n)/A084127(n).