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

1,1

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, d<n} d!. What is the smallest counterexample, if there is any? - M. F. Hasler, Dec 30 2016

Links

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

Example

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.

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 *)

Prog

(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, d<n} d!) for terms of the sequence. - M. F. Hasler, Dec 30 2016

Crossrefs

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

Sequence in context: A077969 A122181 A167758 * A075422 A230195 A098030

Adjacent sequences: A248690 A248691 A248692 * A248694 A248695 A248696

Keyword

nonn,easy

Author

Clark Kimberling, Oct 12 2014

Status

approved