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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A007602, A255725, A255726.

Sequence in context: A146895 A300008 A068343 * A046334 A046382 A168662

Adjacent sequences: A256515 A256516 A256517 * A256519 A256520 A256521

Keyword

nonn,base

Author

Paolo P. Lava, Apr 01 2015

Status

approved