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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.

%I A380995 #64 May 01 2025 21:40:27
%S A380995 385,455,595,1001,1309,1463,1547,1729,1771,2093,2233,2261,2387,2431,
%T A380995 2717,3289,3553,4147,4199,4301,4433,4807,5083,5291,5423,5681,5797,
%U A380995 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
%N 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.
%C A380995 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).
%C A380995 Sequence is intersection of A007304 (sphenics) and A088382 (numbers not exceeding the 4th power of their smallest prime factor).
%C A380995 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.
%C A380995 A380438 is the 'less strong' sequence of sphenic numbers k = p*q*r, where k^(1/5) < p < q < r.
%H A380995 David A. Corneth, <a href="/A380995/b380995.txt">Table of n, a(n) for n = 1..10000</a> (first 231 terms from Matthew Goers)
%H A380995 Sh. T. Ishmukhametov and F. F. Sharifullina, <a href="http://kpfu.ru/portal/docs/F1021095055/e05_08.pdf">On distribution of semiprime numbers</a>, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. <a href="https://doi.org/10.3103/S1066369X14080052">On distribution of semiprime numbers</a>, English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, <a href="https://www.researchgate.net/publication/272041598_On_distribution_of_semiprime_numbers">ResearchGate link</a>.
%H A380995 Matthew Goers, <a href="/A380995/a380995_2.txt">Factors of Terms</a>
%e A380995 595 = 5*7*17 and 595^(1/4) < 5, so 595 is in the sequence.
%e A380995 665 = 5*7*19 but 665^(1/4) > 5, so 665 is not in the sequence.
%t A380995 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 *)
%o A380995 (Python)
%o A380995 from math import isqrt
%o A380995 from sympy import primepi, primerange, integer_nthroot
%o A380995 def A380995(n):
%o A380995     def bisection(f,kmin=0,kmax=1):
%o A380995         while f(kmax) > kmax: kmax <<= 1
%o A380995         kmin = kmax >> 1		
%o A380995         while kmax-kmin > 1:
%o A380995             kmid = kmax+kmin>>1
%o A380995             if f(kmid) <= kmid:
%o A380995                 kmax = kmid
%o A380995             else:
%o A380995                 kmin = kmid
%o A380995         return kmax
%o A380995     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))
%o A380995     return bisection(f,n,n) # _Chai Wah Wu_, Mar 28 2025
%o A380995 (PARI) is(n) = my(f = factor(n)); f[,2] == [1,1,1]~ && f[1,1]^4 > n \\ _David A. Corneth_, Apr 24 2025
%Y A380995 Cf. A007304 (sphenics), A088382, A380438, A115957, A362910 (strong semiprimes), A251728, A138109.
%Y A380995 Subsequence of A253567, A290965.
%K A380995 nonn
%O A380995 1,1
%A A380995 _Matthew Goers_, Feb 12 2025