A060737 -id:A060737 - cp's OEIS frontend

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

Labos Elemer, Apr 23 2001

nonn

			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.

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

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

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

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

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

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

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

A060738 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n! which are not also divisors of n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 4, 5, 12, 51, 92, 225, 340, 573, 1325, 2334, 6024, 8724, 13130, 21601, 46169, 67213, 106427, 178014, 242104, 338499, 727248, 988029, 1924615, 2426894, 3592164, 5817845, 8360196, 10396523, 21941765, 33649653, 48804040, 61413482, 124029358

Offset: 0

Labos Elemer, Apr 25 2001

nonn

			For n up to 7 all divisor differences of n! are also divisors of n!.
For n = 8, there are 3 divisor differences of 8! = 40320 which are not divisors of 8!, namely 27, 54 and 108.

Cf. A000142, A027423, A060737, A060742, A060763.

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

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

a(n) = A060763(n!).

Edited by Dean Hickerson, Jan 22 2002
More terms from Sean A. Irvine, Dec 21 2022
a(41) from Amiram Eldar, Jun 15 2024

cp's OEIS Frontend

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

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