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A096091 Numbers n with property that largest number formed from digits of n (A004186(n)) is divisible by smallest number formed from digits of n (A004185(n)).

%I A096091 #32 Jul 14 2019 21:55:35
%S A096091 1,2,3,4,5,6,7,8,9,10,11,20,22,30,33,40,44,50,55,60,66,70,77,80,88,90,
%T A096091 99,100,101,105,108,110,111,150,180,200,202,220,222,300,303,330,333,
%U A096091 400,404,405,440,444,450,500,501,504,505,510,540,550,555,600,606
%N A096091 Numbers n with property that largest number formed from digits of n (A004186(n)) is divisible by smallest number formed from digits of n (A004185(n)).
%C A096091 The number N = d*10^m*(10^k-1)/9 is a member for all m, k where 1 < d < 10, since the quotient is 10^m. E.g., for d = 7, m = 4, k = 8 we get N = 777777770000.
%C A096091 Conjecture: There are infinitely many terms besides these.
%C A096091 From _Jon E. Schoenfield_, Jul 26 2015: (Start)
%C A096091 Every number whose nonzero digits are all identical (e.g., 70770070777) is a term in the sequence (so the sequence is infinite). Also, if k is a term, then so is k*10 (hence, so is k*10^m for m >= 1). Removal of all terms that satisfy either of the above criteria still leaves an infinite number of terms, beginning with 105, 108, 150, 180, 405, 450, 501, 504, 510, 540, 801, 810, ... (see A260461).
%C A096091 If any integer k is a term, then so is every integer obtained by permuting the digits of k, except for some (not necessarily all) permutations beginning with 0. E.g., since 12000 is a member, so are all the other permutations of its digits that begin with 1 (i.e., 10002, 10020, and 10200), and all those that begin with 2 (i.e., 20001, 20010, 20100, and 21000), as well as the permutations that begin with a single 0 (which, after leading zeros are removed, reduce to 1002, 1020, 1200, 2001, 2010, and 2100), but not those that begin with more than one 0 (i.e., the sequence does not include 12, 21, 102, 120, 201, or 210). Aside from those terms whose nonzero digits are all identical, it appears that only a small number of patterns result from sorting the digits in increasing order (and discarding the zeros, which, of course, are all leading zeros): these "primitives" begin with 12, 15, 16, 18, 24, 25, 36, 45, 48, 125, ... (see A260462). (End)
%H A096091 Giovanni Resta, <a href="/A096091/b096091.txt">Table of n, a(n) for n = 1..10000</a>
%e A096091 110 is a member as 110/011=10.
%p A096091 isA096091 := proc(n)
%p A096091     if modp( A004186(n),A004185(n))= 0 then
%p A096091         true;
%p A096091     else
%p A096091         false;
%p A096091     end if;
%p A096091 end proc:
%p A096091 for n from 1 to 1000 do
%p A096091     if isA096091(n) then
%p A096091         printf("%d,",n) ;
%p A096091     end if;
%p A096091 end do: # _R. J. Mathar_, Jul 26 2015
%t A096091 Select[Range[999], (d = Sort@ IntegerDigits@ #; Divisible@@ FromDigits/@ {Reverse@ d, d})&] (* _Giovanni Resta_, Jul 26 2015 *)
%Y A096091 Cf. A004185, A004186, A096089, A096090, A260461, A260462.
%K A096091 base,nonn
%O A096091 1,2
%A A096091 _Amarnath Murthy_, Jun 22 2004
%E A096091 Edited, corrected and extended by _Robert G. Wilson v_, Jun 23 2004
%E A096091 Entry revised by Editors of the OEIS, Jul 26 2015