A382022 Composite integers k = p*q*r where p < q < r are distinct primes such that p*r < q^2.

70, 105, 110, 154, 182, 231, 238, 266, 273, 286, 322, 374, 418, 429, 442, 494, 506, 561, 598, 627, 638, 646, 663, 682, 715, 741, 754, 759, 782, 806, 814, 874, 897, 902, 935, 946, 957, 962, 969, 986, 1001, 1023, 1034, 1045, 1054, 1066, 1102, 1105, 1118

Offset: 1

Author

Matthew Goers, Mar 12 2025

Keywords

• nonn

Comments

These are squarefree, 3-almost primes, called sphenic numbers, that are less than the cube of the middle prime factor. If k = p*q*r and p < q < r, it is always true that p^3 < k < r^3. This sequence includes the terms where k < q^3.

Examples

			70 = 2*5*7 and 2*7 < 5^2, so 70 is in the sequence.
105 = 3*5*7 and 3*7 < 5^2, so 105 is in the sequence.
165 = 3*5*11 but 3*11 > 5^2, so 165 is not in the sequence.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1776 from Matthew Goers)
• Matthew Goers, Factors of Terms

Crossrefs

Subsequence of A007304 (sphenic numbers).

Supersequence of A375008 (consecutive primes p, q, r).

Cf. A381736.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]] * f[[3, 1]] < f[[2, 1]]^2]; Select[Range[1200], q] (* Amiram Eldar, Mar 12 2025 *)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A382022(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 n+x-sum(primepi(min(x//(p*q),q**2//p))-b for a,p in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,q in enumerate(primerange(p+1,isqrt(x//p)+1),a+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025