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

1,1

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

Links

Table of n, a(n) for n=1..57.

Example

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.

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 *)

Crossrefs

Cf. A137428.

Sequence in context: A083348 A174261 A063539 * A081925 A049199 A337806

Adjacent sequences: A253293 A253294 A253295 * A253297 A253298 A253299

Keyword

nonn

Author

Michael Savoric, Dec 30 2014

Status

approved