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A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = p*q*r, where p, q, r are primes and k^(1/5) < p < q < r.

%I A380438 #63 May 27 2025 08:11:42
%S A380438 30,105,165,195,231,385,455,595,665,715,805,935,1001,1015,1045,1085,
%T A380438 1105,1235,1265,1295,1309,1435,1463,1495,1505,1547,1595,1615,1645,
%U A380438 1705,1729,1771,1855,1885,1955,2015,2035,2065,2093,2135,2185,2233,2255,2261,2345,2365,2387,2405,2431,2465,2485
%N A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = p*q*r, where p, q, r are primes and k^(1/5) < p < q < r.
%C A380438 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.
%C A380438 The only even term is 30 = 2*3*5.
%C A380438 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
%H A380438 Chai Wah Wu, <a href="/A380438/b380438.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..420 from Matthew Goers).
%H A380438 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>.
%e A380438 231 = 3*7*11 and 231^(1/5) < 3, so 231 is in the sequence.
%e A380438 255 = 3*5*17 but 255^(1/5) > 3, so 255 is not in the sequence.
%t A380438 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 *)
%o A380438 (PARI) isok(k) = my(f=factor(k)); (bigomega(f)==3) && (omega(f)==3) && (k < vecmin(f[,1])^5); \\ _Michel Marcus_, Jan 27 2025
%o A380438 (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
%o A380438 (Python)
%o A380438 from math import isqrt
%o A380438 from sympy import primepi, primerange, integer_nthroot
%o A380438 def A380438(n):
%o A380438     def bisection(f,kmin=0,kmax=1):
%o A380438         while f(kmax) > kmax: kmax <<= 1
%o A380438         kmin = kmax >> 1
%o A380438         while kmax-kmin > 1:
%o A380438             kmid = kmax+kmin>>1
%o A380438             if f(kmid) <= kmid:
%o A380438                 kmax = kmid
%o A380438             else:
%o A380438                 kmin = kmid
%o A380438         return kmax
%o A380438     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))
%o A380438     return bisection(f,n,n) # _Chai Wah Wu_, Mar 28 2025
%Y A380438 Subsequence of A253567, A290965, A379271, and A007304.
%Y A380438 A046301 is a subsequence (product of 3 successive primes).
%Y A380438 Cf. A115957, A138109, A251728, A362910 (strong semiprimes), A380995.
%K A380438 nonn
%O A380438 1,1
%A A380438 _Matthew Goers_, Jan 24 2025