A248693 -id:A248693 - 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

A248694 Odd numbers k such that the product of factorials of proper divisors of k does not divide k!

Original entry on oeis.org

1575, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7875, 8085, 8505, 9135, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955, 18585, 19215, 19305, 19635, 19845, 20475, 21105

Offset: 1

Clark Kimberling, Oct 12 2014

Is every term a multiple of 15?

			Let Q(n) = n!/(product of the proper divisors of n).  Then Q(a(1)) = N/D, where
N = 5684447784292091153753490743683678065401858735042447947039489051000812450375
96097513516069953368913418355291373745171485235214678313832004750769512243570213
17329337171352254034549578081676132223379527381310854584384222707565139611863694
30640235947065824459529570708496537565356400158201705547084883398448628433876851
45027005261482010735089664203432206284363981438356549492398792517845833981953470
11656242198592638046634626564224371702079154967385329347508423182940832991802020
79729029554388862293013132314260080936743113826499242289158506713167932372780456
35478273889560173970236977657583276028820238215910913046229649347260976100815721
30514647760639026540100704923229175316918713642498606000080175020651282466288262
98949351261557660288695788306642653729696037113284350082653527072196917361726113
54899425631403773739526421814643000628257393623101868506322157155784868034570876
94944795541518746296597484128788416505102809202721810698024533196028905603665016
69160279455786632925907732365477717222525851009413148227878461914230042744382703
86503762563952079367594382974183048359869258795095100225724624706161418465202017
63929456631534509078212894380900186189087478989553793461391475049200947755236771
281068356116704064311250402286679067
and
D = 6449725440000.

Cf. A248692, A027750.

d[n_] := Drop[Divisors[n], -1]!; p[n_] := Apply[Times, d[n]]
u = Select[Range[25000], ! IntegerQ[#!/p[#]] &]; (* A248693 *)
Select[u, OddQ[#] &]  (* A248694 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions