A060742 Number of divisors of n! which are also differences between consecutive divisors of n! (ordered by size).

0, 0, 1, 2, 4, 9, 15, 27, 41, 68, 111, 218, 328, 624, 929, 1518, 2016, 3689, 4965, 9252, 13177, 20016, 30697, 56749, 69434, 94242, 149558, 190292, 258370, 492924, 615063, 1149403, 1325124, 1841343, 2737190, 3592273, 4193855, 8216492, 12668800, 17654339, 20368544

Offset: 0

Author

Labos Elemer, Apr 23 2001

Keywords

• nonn

Examples

			For n = 5, n! = 120; divisors = {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}; differences = {1,1,1,1,1,2,2,2,3,5,4,6,10,20,60}; intersection = {1,2,3,4,5,6,10,20,60}, so a(5) = 9.
		

Links

• Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.

Crossrefs

Cf. A000142, A027423, A060737, A060738, A060741.

Programs

Maple

f:= proc(n) local D,L;
  D:= numtheory:-divisors(n!);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) intersect D);
end proc:
map(f, [$0..34]); # Robert Israel, Jul 03 2017

Mathematica

a[n_ ] := Length[Intersection[Drop[d=Divisors[n! ], 1]-Drop[d, -1], d]]

PARI

a(n) = {my(v = List(), f = n!, d1 = 1, del); fordiv(f, d, if(d > 1, del = d - d1; if(!(f % del), listput(v, del)); d1 = d)); #Set(v);} \\ Amiram Eldar, Jun 15 2024

Formula

a(n) = A060741(n!/2) for n >= 2. - Amiram Eldar, Jun 15 2024

Extensions

Edited by Dean Hickerson, Jan 22 2002

One more term from Robert G. Wilson v, Jan 29 2002

a(33)-a(35) from Robert Israel, Jul 03 2017

a(36)-a(40) from Amiram Eldar, Jun 15 2024