A362910 Semiprimes p*q for which p <= q < p^3.

4, 6, 9, 10, 14, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57, 65, 69, 77, 85, 91, 95, 115, 119, 121, 133, 143, 145, 155, 161, 169, 185, 187, 203, 205, 209, 215, 217, 221, 235, 247, 253, 259, 265, 287, 289, 295, 299, 301, 305, 319, 323, 329, 335, 341, 355, 361, 365

Offset: 1

Author

Alain Rocchelli, May 10 2023

Keywords

• nonn

Links

• Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, alternative link.

Crossrefs

Cf. A001248 (subsequence), A001358, A251728.

Programs

Maple

with(numtheory):
q:= n-> bigomega(n)=2 and (s-> max(s)Alois P. Heinz, May 10 2023

Mathematica

Select[Range[335], (f = FactorInteger[#])[[;; , 2]] == {2} || (f[[;; , 2]] == {1, 1} && f[[2, 1]] < f[[1, 1]]^3) &] (* Amiram Eldar, May 10 2023 *)

PARI

isok(n)=if(bigomega(n)<>2, 0, my(minfact=factor(n)[1,1], maxfact=n/minfact); maxfact
				

Python

from math import isqrt
from sympy import primepi, primerange
def A362910(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(min(x//p,p**3)) for p in primerange(s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 05 2025

Formula

Limit_{n->oo} n*log(a(n))/a(n) = log(3).