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

%I A248693 #26 Feb 27 2020 20:14:38
%S A248693 24,30,36,40,48,54,60,72,80,84,90,96,100,102,108,112,120,126,132,140,
%T A248693 144,150,156,160,162,168,176,180,192,198,200,204,208,210,216,220,224,
%U A248693 228,234,240,252,260,264,270,272,276,280,288,294,300,306,308,312,320
%N A248693 Numbers k such that the product of factorials of proper divisors of k does not divide k!.
%C A248693 The least odd k in this sequence is 1575; see A248694 and A075460.
%C A248693 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
%H A248693 Clark Kimberling, <a href="/A248693/b248693.txt">Table of n, a(n) for n = 1..1000</a>
%e A248693 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.
%t A248693 d[n_] := Drop[Divisors[n], -1]!; p[n_] := Apply[Times, d[n]]
%t A248693 u = Select[Range[25000], ! IntegerQ[#!/p[#]] &]; (* A248693 *)
%t A248693 Select[u, OddQ[#] &]  (* A248694 *)
%t A248693 (* Second program *)
%t A248693 Select[Range@ 320, ! Divisible[#!, Times @@ Map[Factorial, Most@ Divisors@ #]] &] (* _Michael De Vlieger_, Dec 31 2016 *)
%o A248693 (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
%Y A248693 Cf. A075422 (primitive terms = not a multiple of an earlier term), A248694 (odd terms), A075460 (odd primitive terms), A075071, A027750.
%K A248693 nonn,easy
%O A248693 1,1
%A A248693 _Clark Kimberling_, Oct 12 2014