A255726 Numbers n = concat(x,y) such that the product x*y | n. No leading zeros in y allowed.

11, 12, 15, 24, 36, 110, 120, 125, 150, 240, 315, 360, 735, 1100, 1125, 1200, 1250, 1352, 1500, 1734, 2400, 3150, 3375, 3600, 7350, 11000, 11250, 12000, 12500, 13520, 14112, 15000, 17340, 18144, 21168, 24000, 31500, 33750, 36000, 42336, 63504, 67335, 73500, 91125

Offset: 1

Author

Paolo P. Lava, Apr 01 2015

Keywords

• nonn
• base

Comments

Subset of A255725.

Values of the ratio n / (x*y) are 2, 3, 5, 6, 7, 9 and 11.

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

			11 = concat(1,1); 1*1 = 1 and 11 / 1 = 11.
12 = concat(3,6); 1*2 = 2 and 12 / 2 = 6.
240 = concat(2,40); 2*40 = 80 and 240 / 80 = 3.
		

Links

• Paolo P. Lava and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from Paolo P. Lava)

Crossrefs

Cf. A007602, A255725, 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 i=ilog10(b)+1 then
if a*b>0 then if type(n/(a*b),integer) then print(n);
fi; 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 && 10^(e-1) <= 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 *)