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

Author

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

Keywords

• nonn

Comments

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.

Examples

			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
		

Crossrefs

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).

Programs

Mathematica

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

Formula

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

Extensions

Edited by M. F. Hasler, Dec 30 2016