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.

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

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 * A369353 A088493 A074451

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