cp's OEIS Frontend

A084687 Nontrivial numbers k containing no zero digits which are divisible by the number formed by writing the digits of k in ascending order.

%I A084687 #13 Jun 08 2025 16:15:42
%S A084687 9513,81816,93513,94143,95193,816816,888216,933513,934143,935193,
%T A084687 941493,951993,2491578,8166816,8868216,9333513,9334143,9335193,
%U A084687 9341493,9351993,9414993,9519993,24915798,49827156,81666816,87127446,88668216,93333513
%N A084687 Nontrivial numbers k containing no zero digits which are divisible by the number formed by writing the digits of k in ascending order.
%C A084687 Sequence excludes numbers which are already sorted, like 1234 or 133778, as sorting any such number yields the same number, which is of course divisible by itself, a trivial case.
%C A084687 All members of this sequence appear to be divisible by 3. Further, many of the terms of the sequence can be generated patternistically simply by inserting digits in certain places in earlier terms. The primitive terms of this sequence which cannot be patternistically generated are in sequence A086083.
%C A084687 All terms are divisible by 3.  Proof: if the digits of x*y are a permutation of the digits of x, we must have x*y==x (mod 9), implying either x == 0 (mod 3) or y == 1 (mod 9). - _Robert Israel_, Jul 09 2020
%H A084687 Robert Israel, <a href="/A084687/b084687.txt">Table of n, a(n) for n = 1..208</a>
%e A084687 9513/1359 = 7; 9876543192/1234567899 = 8; etc.
%p A084687 S:= [seq([i],i=1..9)]: R:= NULL: count:= 0:
%p A084687 for d from 2 to 8 do
%p A084687   S:= map(t -> seq([i,op(t)],i=1..t[1]), S);
%p A084687   for s in S do
%p A084687     x:= add(s[i]*10^(d-i),i=1..d);
%p A084687     if x mod 3 <> 0 then next fi;
%p A084687     for m from 2 to 10^(d+1)/x do
%p A084687       if sort(convert(m*x,base,10))=s then
%p A084687         count:= count+1; R:= R, m*x;
%p A084687       fi
%p A084687     od
%p A084687   od
%p A084687 od:
%p A084687 sort([R]); # _Robert Israel_, Jul 09 2020
%t A084687 Select[ Range[ 10^8], IntegerQ[ # /FromDigits[ Sort[ IntegerDigits[ # ]]]] && # != FromDigits[ Sort[ IntegerDigits[ # ]]] && Count[ IntegerDigits[ # ], 0] == 0 & ]
%Y A084687 Cf. A086083.
%K A084687 base,nonn
%O A084687 1,1
%A A084687 _Chuck Seggelin_, Jun 30 2003
%E A084687 Edited by _Robert G. Wilson v_, Jul 07 2003