

A248693


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


3



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
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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 2adic valuation, and P = product_{dn, 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, n1, i, Mod(n, prod(j=2, 1+#n=divisors(n), n[j]!))) \\ Returns nonzero (actually, Mod(n!, P) where P = product_{dn, 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,changed


AUTHOR

Clark Kimberling, Oct 12 2014


STATUS

approved



