A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 42, 48, 54, 60, 64, 72, 96, 100, 108, 120, 128, 144, 156, 162, 168, 180, 192, 216, 240, 256, 272, 288, 294, 300, 324, 342, 360, 384, 432, 480, 486, 500, 504, 512, 576, 600, 648, 720, 768, 840, 900, 960, 972, 1008, 1024

Offset: 1

Author

Labos Elemer, Apr 24 2001

Keywords

• nonn

Comments

Equivalently, A060763(n)=0.

Powers of 2 and factorials up to 7! are here.

For each k=1..A000005(a(n))-1 exists k' < A000005(a(n)) such that A193829(a(n),k) = A027750(a(n),k'). - Reinhard Zumkeller, Jun 25 2015

From Robert Israel, Jul 03 2017: (Start)

Also includes 3*2^k and 2*3^k for all k>= 1.

All terms except 1 are even. (End)

Conjecture: a(n) has the property that for each prime divisor p, p-1|a(n)/p. If this conjecture is true then terms can be searched by distinct prime divisors. - David A. Corneth, Jul 06 2017

The divisors of a(n) form a Brauer chain. See A079301 for the definition of a Brauer chain. - Zizheng Fang, Jan 30 2020

Examples

			For n = 12, divisors={1, 2, 3, 4, 6, 12}; differences={1, 1, 1, 2, 6}; every difference is a divisor, so 12 is in the sequence.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..750

Crossrefs

Cf. A060683, A060763, A193829, A027750.

Programs

Haskell

import Data.List (sort, nub); import Data.List.Ordered (subset)
a060765 n = a060765_list !! (n-1)
a060765_list = filter
(\x -> sort (nub $ a193829_row x) `subset` a027750_row' x) [1..]
-- Reinhard Zumkeller, Jun 25 2015

Magma

[k:k in [1..1025]| forall{i:i in [2..#Divisors(k)]|k mod (d[i]-d[i-1]) eq 0 where d is Divisors(k)}]; // Marius A. Burtea, Jan 30 2020

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) minus D);
end proc:
select(f=0, [$1..1000]); # Robert Israel, Jul 03 2017

Mathematica

test[n_ ] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]==0; Select[Range[1, 1024], test]
(* Second program: *)
Select[Range[2^10], Function[n, AllTrue[Differences@ Divisors@ n, Divisible[n, #] &]]] (* Michael De Vlieger, Jul 12 2017 *)

PARI

isok(n)=my(d=divisors(n), v=vecsort(vector(#d-1, k, d[k+1]-d[k]),,8)); #select(x->setsearch(d, x), v) == #v; \\ Michel Marcus, Jul 06 2017

PARI

is(n)=my(t); fordiv(n,d, if(n%(d-t), return(0)); t=d); 1 \\ Charles R Greathouse IV, Jul 12 2017

Extensions

Edited by Dean Hickerson, Jan 22 2002