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

1,1

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

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).

Mathematica

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

Prog

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

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.

Sequence in context: A066072 A239586 A180611 * A377214 A089934 A113460

Adjacent sequences: A084124 A084125 A084126 * A084128 A084129 A084130

Keyword

nonn

Author

Reinhard Zumkeller, May 15 2003

Extensions

Corrected by T. D. Noe, Nov 15 2006

Status

approved