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"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).
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%I #26 Jun 11 2020 11:54:10

%S 60,84,120,132,168,180,210,240,252,264,280,300,312,315,330,336,360,

%T 390,396,408,420,440,456,468,480,495,504,510,520,528,540,552,560,570,

%U 585,588,600,612,616,624,630,660,672,680,684,690,693,720,728,756,760,765,770,780

%N "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 These form a subsemigroup and a subsequence of the sequence A289484.

%C 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 60=2^2*3*5 is a term because 2^2 > 3 and 2^2*3 > 5.

%e 315=3^2*5*7 is a term because 3^2 > 5 and 3^2*5 > 7.

%p isA291125 := proc(n)

%p local pset,p,pprodidx,pprod,nu,falls ;

%p pset := sort(convert(numtheory[factorset](n),list)) ;

%p pprod := 1;

%p falls := 0 ;

%p for pprodidx from 1 to nops(pset)-1 do

%p p := pset[pprodidx] ;

%p nu := padic[ordp](n,p) ;

%p pprod := pprod*p^nu ;

%p if pprod > pset[pprodidx+1] then

%p falls := falls+1 ;

%p if falls >= 2 then

%p return true;

%p end if;

%p end if;

%p end do:

%p return false ;

%p end proc:

%p for n from 1 to 3000 do

%p if isA291125(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Oct 20 2017

%o (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 Cf. A289484.

%K nonn

%O 1,1

%A _Richard Locke Peterson_, Aug 17 2017

%E New name from _Charles R Greathouse IV_, Jun 11 2020