OFFSET
1,1
COMMENTS
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 12 2014
STATUS
approved