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
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
KEYWORD
nonn
AUTHOR
Michael Savoric, Dec 30 2014
STATUS
approved