A075460 -id:A075460 - cp's OEIS frontend

A075422 Primitive numbers n such that the product of factorials of all proper divisors of n does not divide n!.

24, 30, 36, 40, 54, 84, 100, 102, 112, 126, 132, 140, 156, 176, 198, 208, 220, 228, 234, 260, 272, 276, 294, 308, 340, 342, 348, 350, 364, 372, 380, 392, 414, 444, 460, 462, 476, 490, 492, 516, 522, 532, 546, 558, 564, 572, 580, 608, 620, 636, 644, 666, 708

Offset: 1

Robert G. Wilson v & Amarnath Murthy, Sep 14 2002

nonn

If a number is in the sequence, all of its multiples also meet the criterion, but are not included. This is what the word "primitive" refers to.

			The product of the factorials of the proper divisors of 24, 1! * 2! * 3! * 4! * 6! * 8! * 12!, is divisible by 2^26 and therefore does not divide 24! (which is divisible by 2^22 only). 24 is the smallest number with this property. - _M. F. Hasler_, Dec 31 2016

Cf. A075071. See A075460 for the odd terms of this sequence.

See A248693 for the list of all (also non-primitive) terms (and PARI code).

f[n_] := n!/Apply[Times, Drop[Divisors[n], -1]! ]; a = {}; Do[b = f[n]; If[ !IntegerQ[b], If[ Select[n/a, IntegerQ] == {}, a = Append[a, n]]], {n, 1, 725}]; a

a(n) appears to be asymptotic to c*n with 12 < c < 15. - Benoit Cloitre, Sep 16 2002

Edited by M. F. Hasler, Dec 30 2016

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

Clark Kimberling, Oct 12 2014

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, dM. F. Hasler, Dec 30 2016

			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.

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

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

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

cp's OEIS Frontend

A075422 Primitive numbers n such that the product of factorials of all proper divisors of n does not divide n!.

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