A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703

Offset: 1

Author

David S. Newman, May 04 2008

Keywords

• nonn

Comments

This sequence was suggested by Moshe Shmuel Newman.

A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014

In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017

If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022

Examples

			6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
From _Michael De Vlieger_, Apr 27 2024: (Start):
Table of p*q where p = prime(n) and q = prime(n+k):
n\k   1     2     3     4     5     6     7     8     9    10    11
-------------------------------------------------------------------
1:    6;
2:   15,   21;
3:   35,   55,   65,   85,   95,  115;
4:   77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329;
     ... (End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A251728 and of A006881. A006094 is a proper subset.

Cf. A020639, A010846, A079047, A162306.

Programs

Haskell

a138109 n = a138109_list !! (n-1)
a138109_list = filter f [1..] where
   f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
-- Reinhard Zumkeller, Dec 17 2014

Mathematica

s = {}; Do[f = FactorInteger[i]; test = f[[1]][[1]]; If [test < N[i^(1/2)] && test > N[i^(1/3)], s = Union[s, {i}]], {i, 2, 2000}]; Print[s]
Select[Range[1000],Surd[#,3]Harvey P. Dale, May 10 2015 *)

PARI

is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ Charles R Greathouse IV, Mar 28 2017

PARI

list(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3,sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2,lim\p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Mar 28 2017

Python

from math import isqrt
from sympy import primepi, primerange
def A138109(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*(t-1)>>1)-sum(primepi(min(x//p,p**2)) for p in primerange(s+1)))
    return bisection(f,n,n) # Chai Wah Wu, Mar 05 2025

Formula

From Michael De Vlieger, Apr 27 2024: (Start)

Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.

A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)