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 defines sphenic numbers with an analogous 'strength' as a product of 3 distinct primes k = p*q*r where each prime is greater than k^(1/5), or, alternately, k < p^5.
The only even term is 30 = 2*3*5.
As there are many equivalent ways of expressing Ishmukhametov and Sharifullina's "strongly semiprime" criterion, it is not obvious how it should most appropriately be extended to measure an equivalent "strength" of numbers with more prime factors. Here we follow a comparison of the least prime factor, p, to the factored number, k; but we could instead compare the greatest prime factor, r, to k; or p to r; or measure the variance/standard deviation of the prime factors (more precisely, after twice taking the logarithm of each factor as is done in A379271). Furthermore, it looks clear that the comparison used here (p against k^(1/5)) could be shown to give a substantially lower density asymptotically within the sphenics than Ishmukhametov and Sharifullina's equivalent for semiprimes. - Peter Munn, Feb 18 2025 and May 13 2025
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..420 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.
EXAMPLE
231 = 3*7*11 and 231^(1/5) < 3, so 231 is in the sequence.
255 = 3*5*17 but 255^(1/5) > 3, so 255 is not in the sequence.
MATHEMATICA
q[k_] := Module[{f = FactorInteger[k]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]^5 > k]; Select[Range[2500], q] (* Amiram Eldar, Feb 14 2025 *)
PROG
(PARI) isok(k) = my(f=factor(k)); (bigomega(f)==3) && (omega(f)==3) && (k < vecmin(f[, 1])^5); \\ Michel Marcus, Jan 27 2025
(PARI) list(lim)=my(v=List()); forprime(p=2, sqrtnint(lim\=1, 3), forprime(q=p+1, min(sqrtint(lim\p), p^2), forprime(r=q+2, min(lim\(p*q), p^4\q), listput(v, p*q*r)))); Set(v) \\ Charles R Greathouse IV, May 20 2025
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A380438(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**4//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
CROSSREFS
A046301 is a subsequence (product of 3 successive primes).
KEYWORD
nonn
AUTHOR
Matthew Goers, Jan 24 2025
STATUS
approved