A084127 Prime factor >= other prime factor of n-th semiprime.

2, 3, 3, 5, 7, 5, 7, 11, 5, 13, 11, 17, 7, 19, 13, 23, 7, 17, 11, 19, 29, 31, 13, 23, 37, 11, 41, 17, 43, 29, 13, 31, 47, 19, 53, 37, 23, 59, 17, 11, 61, 41, 43, 19, 67, 47, 71, 13, 29, 73, 31, 79, 53, 23, 83, 13, 59, 89, 61, 37, 17, 97, 67, 101, 29, 41, 103, 19, 71, 107, 43, 31

Offset: 1

Author

Reinhard Zumkeller, May 15 2003

Keywords

• nonn

Comments

Largest nontrivial divisor of n-th semiprime. [Juri-Stepan Gerasimov, Apr 18 2010]

Greater 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), A084126 (lesser of the prime factors of the semiprimes).

Cf. A014673, A061299, A068318, A087718, A087794, A089994, A089995, A096932, A106550, A106554, A108542, A126663, A131284, A138510, A138511.

Programs

Haskell

a084127 = a006530 . a001358  -- Reinhard Zumkeller, Nov 25 2012

Mathematica

FactorInteger[#][[-1, 1]]& /@ Select[Range[1000], PrimeOmega[#] == 2&] (* Jean-François Alcover, Nov 17 2021 *)

PARI

lista(nn) = {for (n=2, nn, if (bigomega(n)==2, f = factor(n); print1(f[length(f~),1], ", ")););} \\ Michel Marcus, Jun 05 2013

Python

from math import isqrt
from sympy import primepi, primerange, primefactors
def A084127(n):
    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
    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 max(primefactors(bisection(f,n,n))) # Chai Wah Wu, Oct 23 2024

Formula

a(n) = A006530(A001358(n)).

a(n) = A001358(n)/A020639(A001358(n)). [corrected by Michel Marcus, Jul 18 2020]

a(n) = A001358(n)/A084126(n).

Extensions

Corrected by T. D. Noe, Nov 15 2006