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.

2112, 4224, 11110, 13104, 16128, 17136, 21120, 23184, 27216, 32256, 42240, 70224, 76608, 79632, 92736, 100128, 101101, 101808, 110110, 111100, 111375, 127008, 130104, 131040, 161280, 170170, 171360, 200112, 211200, 231840, 272160, 301125, 322560, 391092, 422400

Offset: 1

Author

Paolo P. Lava, Apr 01 2015

Keywords

• nonn
• base

Comments

If n belongs to the sequence also n*10^k does.

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.

Examples

			Only 2 or 3 different concatenations.
Two different concatenations:
92736 = concat(9*2*736) and 92736 / (9*2*736) = 7;
92736 = concat(92*7*36) and 92736 / (92*7*36) = 4.
Three different concatenations:
23184 = concat(2,3,184) and 23184 / (2*3*184) = 21;
23184 = concat(23,1,84) and 23184 / (23*1*84) = 12;
23184 = concat(23,18,4) and 23184 / (23*18*4) = 14.
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.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A007602, A255725, A255726.

Programs

Maple

with(numtheory); P:=proc(q) local a,ab,b,c,i,j,k,m,n,v,w;
v:=array(1..10,1..3); w:=[];for n from 1 to q do j:=0;
for i from 1 to ilog10(n) do c:=(n mod 10^i); ab:=trunc(n/10^i);
for k from 1 to ilog10(ab) do a:=trunc(ab/10^k); b:=ab-a*10^k;
if a*b*c>0 then if type(n/(a*b*c),integer) then j:=j+1;
w:=sort([a,b,c]); for m from 1 to 3 do v[j,m]:=w[m]; od;
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]
then j:=j-1; break; fi; od; fi; fi; od; od;
if j>1 then print(n); fi; od; end: P(10^9);

Mathematica

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 *)