A380995 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 4th root of k: k = p*q*r, where p, q, r are primes and k^(1/4) < p < q < r.

385, 455, 595, 1001, 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387, 2431, 2717, 3289, 3553, 4147, 4199, 4301, 4433, 4807, 5083, 5291, 5423, 5681, 5797, 5863, 6061, 6149, 6409, 6479, 6721, 6851, 6919, 7163, 7337, 7429, 7579, 7657, 7667, 7733, 7843, 8041, 8177, 8437, 8569, 8671, 8723, 8789, 8987, 9061

Offset: 1

Author

Matthew Goers, Feb 12 2025

Keywords

• nonn

Comments

This subsequence of the sphenics (A007304) is similar to A362910 or A138109 for semiprimes. Ishmukhametov and Sharifullina defined semiprimes n = p*q where each prime is greater than n^(1/4) as strongly semiprime. This sequence lists sphenic numbers that are a product of 3 distinct primes k = p*q*r where each prime is greater than k^(1/4).

Sequence is intersection of A007304 (sphenics) and A088382 (numbers not exceeding the 4th power of their smallest prime factor).

No terms have 2 or 3 as a prime factor, as all sphenic numbers are greater than 2^4 = 16 and all odd sphenic numbers are greater than 3^4 = 81.

A380438 is the 'less strong' sequence of sphenic numbers k = p*q*r, where k^(1/5) < p < q < r.

Examples

			595 = 5*7*17 and 595^(1/4) < 5, so 595 is in the sequence.
665 = 5*7*19 but 665^(1/4) > 5, so 665 is not in the sequence.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 231 terms from Matthew Goers)
• Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. On distribution of semiprime numbers, English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, ResearchGate link.
• Matthew Goers, Factors of Terms

Crossrefs

Cf. A007304 (sphenics), A088382, A380438, A115957, A362910 (strong semiprimes), A251728, A138109.

Subsequence of A253567, A290965.

Programs

Mathematica

q[k_] := Module[{f = FactorInteger[k]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]^4 > k]; Select[Range[10^4], q] (* Amiram Eldar, Feb 14 2025 *)

PARI

is(n) = my(f = factor(n)); f[,2] == [1,1,1]~ && f[1,1]^4 > n \\ David A. Corneth, Apr 24 2025

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A380995(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(max(0,primepi(min(x//(p*q),p**3//q))-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