cp's OEIS Frontend

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

%I A060765 #39 Sep 08 2022 08:45:03
%S A060765 1,2,4,6,8,12,16,18,20,24,32,36,42,48,54,60,64,72,96,100,108,120,128,
%T A060765 144,156,162,168,180,192,216,240,256,272,288,294,300,324,342,360,384,
%U A060765 432,480,486,500,504,512,576,600,648,720,768,840,900,960,972,1008,1024
%N A060765 Numbers n such that every difference between consecutive divisors (ordered by increasing magnitude) of n is also a divisor of n.
%C A060765 Equivalently, A060763(n)=0.
%C A060765 Powers of 2 and factorials up to 7! are here.
%C A060765 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
%C A060765 From _Robert Israel_, Jul 03 2017: (Start)
%C A060765 Also includes 3*2^k and 2*3^k for all k>= 1.
%C A060765 All terms except 1 are even. (End)
%C A060765 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
%C A060765 The divisors of a(n) form a Brauer chain. See A079301 for the definition of a Brauer chain. - _Zizheng Fang_, Jan 30 2020
%H A060765 Donovan Johnson, <a href="/A060765/b060765.txt">Table of n, a(n) for n = 1..750</a>
%e A060765 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.
%p A060765 f:= proc(n) local D,L;
%p A060765   D:= numtheory:-divisors(n);
%p A060765   L:= sort(convert(D,list));
%p A060765   nops(convert(L[2..-1]-L[1..-2],set) minus D);
%p A060765 end proc:
%p A060765 select(f=0, [$1..1000]); # _Robert Israel_, Jul 03 2017
%t A060765 test[n_ ] := Length[Complement[Drop[d=Divisors[n], 1]-Drop[d, -1], d]]==0; Select[Range[1, 1024], test]
%t A060765 (* Second program: *)
%t A060765 Select[Range[2^10], Function[n, AllTrue[Differences@ Divisors@ n, Divisible[n, #] &]]] (* _Michael De Vlieger_, Jul 12 2017 *)
%o A060765 (Haskell)
%o A060765 import Data.List (sort, nub); import Data.List.Ordered (subset)
%o A060765 a060765 n = a060765_list !! (n-1)
%o A060765 a060765_list = filter
%o A060765 (\x -> sort (nub $ a193829_row x) `subset` a027750_row' x) [1..]
%o A060765 -- _Reinhard Zumkeller_, Jun 25 2015
%o A060765 (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
%o A060765 (PARI) is(n)=my(t); fordiv(n,d, if(n%(d-t), return(0)); t=d); 1 \\ _Charles R Greathouse IV_, Jul 12 2017
%o A060765 (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
%Y A060765 Cf. A060683, A060763, A193829, A027750.
%K A060765 nonn
%O A060765 1,2
%A A060765 _Labos Elemer_, Apr 24 2001
%E A060765 Edited by _Dean Hickerson_, Jan 22 2002