A255725 Numbers n = concat(x,y) such that the product x*y | n. Leading zeros in y allowed.

11, 12, 15, 24, 36, 101, 102, 104, 105, 110, 120, 125, 150, 208, 240, 306, 315, 360, 735, 1001, 1002, 1004, 1005, 1008, 1010, 1020, 1025, 1040, 1050, 1100, 1125, 1200, 1250, 1352, 1500, 1734, 2016, 2080, 2400, 3006, 3015, 3024, 3060, 3150, 3375, 3600, 6048, 7007

Offset: 1

Author

Paolo P. Lava, Apr 01 2015

Keywords

• nonn
• base

Comments

There are numbers that present an additional quasi-solution. For instance, consider 26733375: it is in the sequence because 26733375 / (267 * 33375) = 3 but 26733375 / (2673337 * 5) = 2.000000374... is close to being an integer, too.

Other examples:

52116672 / (521 * 16672) = 6 and 52116672 / (5211667 * 2) = 5.000000191...

138911112 / (1389 * 11112) = 9 and 138911112 / (13891111 * 2) = 5.0000000719...

Is there any number that admits two or more different concatenations whose multiplications divide the number itself (no term up to 3*10^9) ?

Examples

			15 = concat(1,5); 1*5 = 5 and 15 / 5 = 3.
36 = concat(3,6); 3*6 = 18 and 36 / 18 = 2.
9072 = concat(9,072); 9*72 = 648 and 9072 / 648 = 14.
		

Links

• Paolo P. Lava and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Paolo P. Lava)

Crossrefs

Cf. A007602, A255726, A256518.

Programs

Maple

with(numtheory); P:=proc(q) local a,b,i,n;
for n from 1 to q do for i from 1 to ilog10(n) do
a:=trunc(n/10^i);  b:=n-a*10^i;
if a*b>0 then if type(n/(a*b),integer) then print(n);
fi; fi; od; od; end: P(10^9);

Mathematica

v[e_]:=Block[{x,y,k}, y+10^e*x /. List@ ToRules@ Reduce[k*x*y ==  x*10^e+y && k>=0 && x>0 && 0 < y < 10^e, {k,x,y}, Integers]]; upto[nd_] := Select[ Union@ Flatten@ Array[v,nd], # < 10^nd &]; upto[10] (* terms < 10^10, Giovanni Resta, May 26 2015 *)