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

0,4

Links

Table of n, a(n) for n=0..40.

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

Formula

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

Example

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.

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]]

Prog

(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

Crossrefs

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

Sequence in context: A157254 A080004 A176915 * A060737 A266647 A085683

Adjacent sequences: A060739 A060740 A060741 * A060743 A060744 A060745

Keyword

nonn

Author

Labos Elemer, Apr 23 2001

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

Status

approved