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A256518 Consider numbers n = concat(x,y,z) such that the product x*y*z | n. Leading zeros in y and z allowed. Sequence lists numbers that admit different concatenations.

%I A256518 #19 Apr 14 2015 10:48:47
%S A256518 2112,4224,11110,13104,16128,17136,21120,23184,27216,32256,42240,
%T A256518 70224,76608,79632,92736,100128,101101,101808,110110,111100,111375,
%U A256518 127008,130104,131040,161280,170170,171360,200112,211200,231840,272160,301125,322560,391092,422400
%N A256518 Consider numbers n = concat(x,y,z) such that the product x*y*z | n. Leading zeros in y and z allowed. Sequence lists numbers that admit different concatenations.
%C A256518 If n belongs to the sequence also n*10^k does.
%C A256518 Two concatenations are considered different when one of them is not a permutation of the other. E.g.: (6,2,22,5) and (6,22,2,5) are not different.
%H A256518 Paolo P. Lava, <a href="/A256518/b256518.txt">Table of n, a(n) for n = 1..100</a>
%e A256518 Only 2 or 3 different concatenations.
%e A256518 Two different concatenations:
%e A256518 92736 = concat(9*2*736) and 92736 / (9*2*736) = 7;
%e A256518 92736 = concat(92*7*36) and 92736 / (92*7*36) = 4.
%e A256518 Three different concatenations:
%e A256518 23184 = concat(2,3,184) and 23184 / (2*3*184) = 21;
%e A256518 23184 = concat(23,1,84) and 23184 / (23*1*84) = 12;
%e A256518 23184 = concat(23,18,4) and 23184 / (23*18*4) = 14.
%e A256518 The six concatenations of 111111 are excluded because they are basically just one: 1*11*111; 1*111*11; 11*1*111; 11*111*1; 111*1*11; 111*11*1 and 111111 / (1*11*111) = 91.
%p A256518 with(numtheory); P:=proc(q) local a,ab,b,c,i,j,k,m,n,v,w;
%p A256518 v:=array(1..10,1..3); w:=[];for n from 1 to q do j:=0;
%p A256518 for i from 1 to ilog10(n) do c:=(n mod 10^i); ab:=trunc(n/10^i);
%p A256518 for k from 1 to ilog10(ab) do a:=trunc(ab/10^k); b:=ab-a*10^k;
%p A256518 if a*b*c>0 then if type(n/(a*b*c),integer) then j:=j+1;
%p A256518 w:=sort([a,b,c]); for m from 1 to 3 do v[j,m]:=w[m]; od;
%p A256518 for m from 1 to j-1 do if v[m,1]=v[j,1] and v[m,2]=v[j,2] and v[m,3]=v[j,3]
%p A256518 then j:=j-1; break; fi; od; fi; fi; od; od;
%p A256518 if j>1 then print(n); fi; od; end: P(10^9);
%t A256518 fQ[n_] := Block[{id = IntegerDigits@ n}, lng = Length@ id; t = Times @@@ Union[Sort /@ Partition[ Flatten@ Table[{FromDigits@ Take[id, {1, i}], FromDigits@ Take[id, {i + 1, j}], FromDigits@ Take[id, {j + 1, lng}]}, {i, 1, lng - 2}, {j, i + 1, lng - 1}], 3]]; Count[IntegerQ /@ (n/t), True] > 1]; k = 100; lst = {}; While[k < 1000001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Apr 09 2015 *)
%Y A256518 Cf. A007602, A255725, A255726.
%K A256518 nonn,base
%O A256518 1,1
%A A256518 _Paolo P. Lava_, Apr 01 2015