A253296 Numbers with more composite divisors than prime divisors such that all the prime divisors are smaller than the composite divisors.

8, 12, 16, 18, 24, 27, 30, 32, 36, 45, 48, 50, 54, 63, 64, 70, 72, 75, 81, 90, 96, 98, 105, 108, 125, 128, 135, 144, 147, 150, 154, 162, 165, 175, 182, 189, 192, 195, 216, 225, 231, 242, 243, 245, 250, 256, 270, 273, 275, 286, 288, 315, 324, 325, 338, 343, 350

Offset: 1

Author

Michael Savoric, Dec 30 2014

Keywords

• nonn

Comments

List of composite numbers with n >= 2 nontrivial divisors where the k smallest nontrivial divisors are all primes and the n - k largest nontrivial divisors are all nonprimes, 1 <= k < n.

Here the term "nontrivial divisors" only serves to exclude 1.

Except for semiprimes, all composite numbers have more composite divisors than prime divisors. - Robert G. Wilson v, Jan 12 2015

Examples

			36 is in the sequence because its nontrivial divisors are 2, 3, 4, 6, 9, 12, 18, and of these, the first two are prime and the rest are composite.
40 is not in the sequence because its nontrivial divisors are 2, 4, 5, 8, 10, 20, and the composite divisor 4 falling between the prime divisors 2 and 5 disqualifies 40 from membership in the sequence.
		

Crossrefs

Cf. A137428.

Programs

Maple

filter:= proc(n)
local f,x;
f:= ifactors(n)[2];
if mul(t[2]+1,t=f) <= 2*nops(f)+1 then return false fi;
if f[1,2] > 1 then x:= f[1,1]^2 else x:= f[1,1]*f[2,1] fi;
max(seq(t[1],t=f)) < x
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 01 2015

Mathematica

ntd[n_] := (dlist = Divisors[n]; dlist[[2 ;; Length[dlist] - 1]])
test[n_] := (tlist = ntd[n];
  If[tlist == {}, False,
   index = 1;
   While[index <= Length[tlist] && PrimeQ[tlist[[index]]] == True,
    index = index + 1];
   If[index == 1 || index > Length[tlist], False,
    While[index <= Length[tlist] && PrimeQ[tlist[[index]]] == False,
     index = index + 1];
    If[index <= Length[tlist], False, True]]])
Select[Table[n, {n, 2, 2500, 1}], test] (* Savoric *)
primeDivs[n_Integer] := Select[Divisors[n], PrimeQ]; compDivs[n_Integer] := Drop[Complement[Divisors[n], primeDivs[n]], 1]; Select[Range[4, 500], Not[PrimeQ[#]] && primeDivs[#][[-1]] < compDivs[#][[1]] && Length[primeDivs[#]] < Length[compDivs[#]] &] (* Alonso del Arte, Dec 31 2014 *)
fQ[n_] := Block[{d = PrimeQ@ Most@ Rest@ Divisors@ n}, d[[1]] == True && d[[-1]] == False && Length@ Split@ d == 2]; Select[ Range@ 350, fQ] (* Robert G. Wilson v, Jan 12 2015 *)