A075071
n! divided by product of factorials of all proper divisors of n, as n runs through the values for which the result is an integer.
Original entry on oeis.org
1, 2, 6, 12, 120, 60, 5040, 840, 60480, 15120, 39916800, 2310, 6227020800, 8648640, 1816214400, 10810800, 355687428096000, 2042040, 121645100408832000, 116396280, 1689515283456000, 14079294028800, 25852016738884976640000
Offset: 1
a(12) = 12! / (2!*3!*4!*6!) = 2310.
-
Select[Table[ n!/ Apply[ Times, Drop[ Divisors[n], -1]! ], {n, 1, 23}], IntegerQ]
Select[Table[n!/Times@@(Most[Divisors[n]]!),{n,30}],IntegerQ] (* Harvey P. Dale, Aug 01 2020 *)
A248693
Numbers k such that the product of factorials of proper divisors of k does not divide k!.
Original entry on oeis.org
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
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.
-
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 *)
-
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
A075460
Odd primitive numbers such that n! divided by product of factorials of all proper divisors of n is not an integer.
Original entry on oeis.org
1575, 2835, 3465, 4095, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 8085, 9135, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 15435, 16695, 18585, 19215, 19635, 21105, 21945, 22275, 22365, 22995, 23205, 24885, 25245, 25935, 26145, 26565
Offset: 1
1575 = 3^2*5^2*7 is in the sequence, because the product of the factorials of its proper divisors { 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525 } does not divide 1575!. (For example, the former's 2-adic valuation equals 1588 while the latter's 2-adic valuation equals only 1569.) This is the smallest odd number with this property. - _M. F. Hasler_, Dec 30 2016
Cf.
A075071. The first primitive n's with this property (most of which are even) are in
A075422.
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f[n_] := n!/Apply[Times, Drop[Divisors[n], -1]! ]; a = {}; Do[b = f[n]; If[ !IntegerQ[b], If[ Select[n/a, IntegerQ] == {}, Print[n]; a = Append[a, n]]], {n, 1, 28213, 2}]; a
Showing 1-3 of 3 results.
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