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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 22 03:20 EST 2019. Contains 329383 sequences. (Running on oeis4.)