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%I #26 Mar 06 2024 04:47:56
%S 16,24,32,36,48,54,60,64,72,80,81,90,96,100,108,112,120,126,128,135,
%T 140,144,150,160,162,168,180,189,192,196,200,210,216,224,225,240,243,
%U 250,252,256,264,270,280,288,294,300,308,315,320,324,330,336,350,352,360
%N A number k is included if both (the largest divisor of k that is <= sqrt(k)) and (the smallest divisor of k that is >= sqrt(k)) are composite.
%C All numbers that are each the square of a composite are included in the sequence. All numbers that are the square of a prime are excluded from the sequence.
%C A prime power p^h is included if and only if h >= 4. - _Robert Israel_, Jun 21 2018
%H Robert Israel, <a href="/A140349/b140349.txt">Table of n, a(n) for n = 1..10000</a>
%e The divisors of 80 are 1,2,4,5,8,10,16,20,40,80. The middle two divisors are 8 and 10, which are both composite. So 80 is included in the sequence.
%p filter:= proc(n) local dp,dm;
%p if issqr(n) then return not isprime(sqrt(n)) fi;
%p dm,dp:= selectremove(t -> t^2 < n, numtheory:-divisors(n));
%p not isprime(max(dm)) and not isprime(min(dp));
%p end proc:
%p select(filter, [$2..1000]); # _Robert Israel_, Jun 21 2018
%t fQ[n_] := Block[{m = DivisorSigma[0, n]}, Union@ PrimeQ@ Take[ Divisors@ n, {Floor[(m + 1)/2], Ceiling[(m + 1)/2]}] == {False}]; Select[ Range[2, 363], fQ@# &] (* _Robert G. Wilson v_, May 31 2008 *)
%t cdQ[n_]:=Module[{d=Divisors[n],a,b},a=Select[d,#<=Sqrt[n]&][[-1]];b= Select[ d,#>=Sqrt[n]&][[1]];AllTrue[{a,b},CompositeQ]]; Select[ Range[ 400],cdQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 17 2019 *)
%Y Cf. A033676, A033677.
%K nonn
%O 1,1
%A _Leroy Quet_, May 29 2008
%E More terms from _Robert G. Wilson v_, May 31 2008
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