A248693 Numbers k such that the product of factorials of proper divisors of k does not divide k!.

24, 30, 36, 40, 48, 54, 60, 72, 80, 84, 90, 96, 100, 102, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 272, 276, 280, 288, 294, 300, 306, 308, 312, 320

Offset: 1

Author

Clark Kimberling, Oct 12 2014

Keywords

• nonn
• easy

Comments

The least odd k in this sequence is 1575; see A248694 and A075460.

It seems that the property is satisfied iff v_2(n!) < v_2(P), where v_2 is the 2-adic valuation, and P = product_{d|n, dM. F. Hasler, Dec 30 2016

Examples

			Let Q(n) = n!/(product of the proper divisors of n).  Then Q(n) an integer for n = 1..23, as indicated by the following list of Q(n) for n = 1..24:  1, 2, 6, 12, 120, 60, 5040, 840, 60480, 15120, 39916800, 2310, 6227020800, 8648640, 1816214400, 10810800, 355687428096000, 2042040, 121645100408832000, 116396280, 1689515283456000, 14079294028800, 25852016738884976640000, 7436429/48.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A075422 (primitive terms = not a multiple of an earlier term), A248694 (odd terms), A075460 (odd primitive terms), A075071, A027750.

Programs

Mathematica

d[n_] := Drop[Divisors[n], -1]!; p[n_] := Apply[Times, d[n]]
u = Select[Range[25000], ! IntegerQ[#!/p[#]] &]; (* A248693 *)
Select[u, OddQ[#] &]  (* A248694 *)
(* Second program *)
Select[Range@ 320, ! Divisible[#!, Times @@ Map[Factorial, Most@ Divisors@ #]] &] (* Michael De Vlieger, Dec 31 2016 *)

PARI

is(n)=prod(i=2,n-1,i,Mod(n,prod(j=2,-1+#n=divisors(n),n[j]!))) \\ Returns nonzero (actually, Mod(n!,P) where P = product_{d|n, dM. F. Hasler, Dec 30 2016