

A140349


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.


1



16, 24, 32, 36, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 126, 128, 135, 140, 144, 150, 160, 162, 168, 180, 189, 192, 196, 200, 210, 216, 224, 225, 240, 243, 250, 252, 256, 264, 270, 280, 288, 294, 300, 308, 315, 320, 324, 330, 336, 350, 352, 360
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OFFSET

1,1


COMMENTS

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.
A prime power p^h is included if and only if h >= 4.  Robert Israel, Jun 21 2018


LINKS

Robert Israel, Table of n, a(n) for n = 1..10000


EXAMPLE

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.


MAPLE

filter:= proc(n) local dp, dm;
if issqr(n) then return not isprime(sqrt(n)) fi;
dm, dp:= selectremove(t > t^2 < n, numtheory:divisors(n));
not isprime(max(dm)) and not isprime(min(dp));
end proc:
select(filter, [$2..1000]); # Robert Israel, Jun 21 2018


MATHEMATICA

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
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 *)


CROSSREFS

Cf. A033676, A033677.
Sequence in context: A320141 A225903 A033987 * A088493 A074451 A253782
Adjacent sequences: A140346 A140347 A140348 * A140350 A140351 A140352


KEYWORD

nonn


AUTHOR

Leroy Quet, May 29 2008


EXTENSIONS

More terms from Robert G. Wilson v, May 31 2008


STATUS

approved



