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

1,1

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.

Links

Table of n, a(n) for n=1..53.

Formula

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

Example

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

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

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).

Sequence in context: A122181 A167758 A248693 * A230195 A098030 A290451

Adjacent sequences: A075419 A075420 A075421 * A075423 A075424 A075425

Keyword

nonn

Author

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

Extensions

Edited by M. F. Hasler, Dec 30 2016

Status

approved