OFFSET
1,1
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.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 231 terms from Matthew Goers)
Matthew Goers, Factors of terms
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.
EXAMPLE
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.
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 *)
PROG
(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
(PARI) is(n) = my(f = factor(n)); f[, 2] == [1, 1, 1]~ && f[1, 1]^4 > n \\ David A. Corneth, Apr 24 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Goers, Feb 12 2025
STATUS
approved