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A291125 "Doubly A289484" numbers: numbers with prime factorization p1^e1 * p2^e2 * ... * pk^ek such that there exist i < j < k with p1^e1 * p2^e2 * ... pi^ei > p(i+1) and p1^e1 * p2^e2 * ... pj^ej > p(j+1).

%I A291125 #26 Jun 11 2020 11:54:10
%S A291125 60,84,120,132,168,180,210,240,252,264,280,300,312,315,330,336,360,
%T A291125 390,396,408,420,440,456,468,480,495,504,510,520,528,540,552,560,570,
%U A291125 585,588,600,612,616,624,630,660,672,680,684,690,693,720,728,756,760,765,770,780
%N A291125 "Doubly A289484" numbers: numbers with prime factorization p1^e1 * p2^e2 * ... * pk^ek such that there exist i < j < k with p1^e1 * p2^e2 * ... pi^ei > p(i+1) and p1^e1 * p2^e2 * ... pj^ej > p(j+1).
%C A291125 These form a subsemigroup and a subsequence of the sequence A289484.
%C A291125 Density: Only 4.3% of the integers between 1 and 400 are doubly A289484.divisible by at least 3 primes. If a term in the sequence is squarefree, it must be divisible by at least 4 primes. If a number n is in the sequence, then every multiple is also in it. Using Wolfram Alpha, about 48% of the integers between 10^40+1 to 10^40+62 were found to be doubly A289484.
%e A291125 60=2^2*3*5 is a term because 2^2 > 3 and 2^2*3 > 5.
%e A291125 315=3^2*5*7 is a term because 3^2 > 5 and 3^2*5 > 7.
%p A291125 isA291125 := proc(n)
%p A291125     local pset,p,pprodidx,pprod,nu,falls ;
%p A291125     pset := sort(convert(numtheory[factorset](n),list)) ;
%p A291125     pprod := 1;
%p A291125     falls := 0 ;
%p A291125     for pprodidx from 1 to nops(pset)-1 do
%p A291125         p := pset[pprodidx] ;
%p A291125         nu := padic[ordp](n,p) ;
%p A291125         pprod := pprod*p^nu ;
%p A291125         if pprod > pset[pprodidx+1] then
%p A291125             falls := falls+1 ;
%p A291125             if falls >= 2 then
%p A291125                 return true;
%p A291125             end if;
%p A291125         end if;
%p A291125     end do:
%p A291125     return false ;
%p A291125 end proc:
%p A291125 for n from 1 to 3000 do
%p A291125     if isA291125(n) then
%p A291125         printf("%d,",n) ;
%p A291125     end if;
%p A291125 end do: # _R. J. Mathar_, Oct 20 2017
%o A291125 (PARI) is(n,f=factor(n))=my(p=1,t,s); for(i=1,#f~, t=f[i,1]^f[i,2]; if(p>t,s++); p*=t); s>1 \\ _Charles R Greathouse IV_, Jun 10 2020
%Y A291125 Cf. A289484.
%K A291125 nonn
%O A291125 1,1
%A A291125 _Richard Locke Peterson_, Aug 17 2017
%E A291125 New name from _Charles R Greathouse IV_, Jun 11 2020