A084126 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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).

MATHEMATICA

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

PROG

(Haskell)

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

(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

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.

Sequence in context: A159953 A074595 A372754 * A136032 A135975 A334796

Adjacent sequences: A084123 A084124 A084125 * A084127 A084128 A084129

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, May 15 2003

STATUS

approved