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Semiprimes p*q for which p <= q < p^3.
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%I #39 May 11 2023 18:48:33

%S 4,6,9,10,14,15,21,25,33,35,39,49,51,55,57,65,69,77,85,91,95,115,119,

%T 121,133,143,145,155,161,169,185,187,203,205,209,215,217,221,235,247,

%U 253,259,265,287,289,295,299,301,305,319,323,329,335,341,355,361,365

%N Semiprimes p*q for which p <= q < p^3.

%C The terms have already been considered as "strongly semiprimes".

%H 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 Limit_{n->oo} n*log(a(n))/a(n) = log(3).

%p with(numtheory):

%p q:= n-> bigomega(n)=2 and (s-> max(s)<min(s)^3)(factorset(n)):

%p select(q, [$4..500])[]; # _Alois P. Heinz_, May 10 2023

%t Select[Range[335], (f = FactorInteger[#])[[;; , 2]] == {2} || (f[[;; , 2]] == {1, 1} && f[[2, 1]] < f[[1, 1]]^3) &] (* _Amiram Eldar_, May 10 2023 *)

%o (PARI) isok(n)=if(bigomega(n)<>2, 0, my(minfact=factor(n)[1,1], maxfact=n/minfact); maxfact<minfact^3)

%o select(isok, [1..500])

%Y Cf. A001248 (subsequence), A001358, A251728.

%K nonn

%O 1,1

%A _Alain Rocchelli_, May 10 2023