cp's OEIS Frontend

A362910 Semiprimes p*q for which p <= q < p^3.

%I A362910 #50 Mar 06 2025 08:29:02
%S A362910 4,6,9,10,14,15,21,25,33,35,39,49,51,55,57,65,69,77,85,91,95,115,119,
%T A362910 121,133,143,145,155,161,169,185,187,203,205,209,215,217,221,235,247,
%U A362910 253,259,265,287,289,295,299,301,305,319,323,329,335,341,355,361,365
%N A362910 Semiprimes p*q for which p <= q < p^3.
%H A362910 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">English translation</a>, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, <a href="https://www.researchgate.net/publication/272041598_On_distribution_of_semiprime_numbers">alternative link</a>.
%F A362910 Limit_{n->oo} n*log(a(n))/a(n) = log(3).
%p A362910 with(numtheory):
%p A362910 q:= n-> bigomega(n)=2 and (s-> max(s)<min(s)^3)(factorset(n)):
%p A362910 select(q, [$4..500])[];  # _Alois P. Heinz_, May 10 2023
%t A362910 Select[Range[335], (f = FactorInteger[#])[[;; , 2]] == {2} || (f[[;; , 2]] == {1, 1} && f[[2, 1]] < f[[1, 1]]^3) &] (* _Amiram Eldar_, May 10 2023 *)
%o A362910 (PARI) isok(n)=if(bigomega(n)<>2, 0, my(minfact=factor(n)[1,1], maxfact=n/minfact); maxfact<minfact^3)
%o A362910 select(isok, [1..500])
%o A362910 (Python)
%o A362910 from math import isqrt
%o A362910 from sympy import primepi, primerange
%o A362910 def A362910(n):
%o A362910     def bisection(f,kmin=0,kmax=1):
%o A362910         while f(kmax) > kmax: kmax <<= 1
%o A362910         kmin = kmax >> 1
%o A362910         while kmax-kmin > 1:
%o A362910             kmid = kmax+kmin>>1
%o A362910             if f(kmid) <= kmid:
%o A362910                 kmax = kmid
%o A362910             else:
%o A362910                 kmin = kmid
%o A362910         return kmax
%o A362910     def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(min(x//p,p**3)) for p in primerange(s+1)))
%o A362910     return bisection(f,n,n) # _Chai Wah Wu_, Mar 05 2025
%Y A362910 Cf. A001248 (subsequence), A001358, A251728.
%K A362910 nonn
%O A362910 1,1
%A A362910 _Alain Rocchelli_, May 10 2023