A060738 -id:A060738 - cp's OEIS frontend

A060763 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n which are not also divisors of n.

0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 3, 1, 1, 0, 2, 1, 3, 1, 1, 1, 1, 0, 3, 1, 3, 0, 1, 1, 3, 1, 1, 0, 1, 1, 4, 1, 1, 0, 2, 2, 3, 1, 1, 0, 3, 2, 3, 1, 1, 0, 1, 1, 4, 0, 3, 1, 1, 1, 3, 3, 1, 0, 1, 1, 3, 1, 3, 1, 1, 2, 4, 1, 1, 1, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 0, 1, 2, 4, 0, 1, 1, 1, 1, 5

Offset: 1

Labos Elemer, Apr 24 2001

nonn

			For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; the differences {3,4,21} are not divisors, so a(70)=3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A060682, A060738, A060764.

f:= proc(n) local D,L;
  D:= numtheory:-divisors(n);
  L:= sort(convert(D,list));
  nops(convert(L[2..-1]-L[1..-2],set) minus D);
end proc:
map(f, [$1..200]); # Robert Israel, Jul 03 2017

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

a(n) = my(d=divisors(n)); #select(x->(setsearch(d, x)==0), vecsort(vector(#d-1, k, d[k+1] - d[k]),,8)); \\ Michel Marcus, Jul 04 2017

Edited by Dean Hickerson, Jan 22 2002

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

Original entry on oeis.org

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

A060737 Number of distinct differences between consecutive divisors of n! (ordered by size).

Original entry on oeis.org

0, 0, 1, 2, 4, 9, 15, 27, 44, 72, 116, 230, 379, 716, 1154, 1858, 2589, 5014, 7299, 15276, 21901, 33146, 52298, 102918, 136647, 200669, 327572, 432396, 596869, 1220172, 1603092, 3074018, 3752018, 5433507, 8555035, 11952469, 14590378, 30158257, 46318453, 66458379

Offset: 0

Labos Elemer, Apr 25 2001

nonn

			For n = 4, n! = 24; divisors = {1,2,3,4,6,8,12,24}; differences = {1,1,1,2,2,4,12}, distinct differences = {1,2,4,12}, so a(4) = 4.

Cf. A000142, A060682, A060738, A060742.

a[n_ ] := Length[Union[Drop[d=Divisors[n! ], 1]-Drop[d, -1]]]
Table[Length[Union[Differences[Divisors[n!]]]],{n,0,40}] (* Harvey P. Dale, Nov 22 2021 *)

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

a(n) = A060682(n!).

Edited by Dean Hickerson, Jan 22 2002
More terms from Ryan Propper, Mar 22 2006
a(37)-a(39) from Amiram Eldar, Jun 15 2024

cp's OEIS Frontend

A060763 Number of distinct differences between consecutive divisors (ordered by increasing magnitude) of n which are not also divisors of n.

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

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Author

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Maple

Mathematica

PARI

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A060737 Number of distinct differences between consecutive divisors of n! (ordered by size).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions