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A255725 Numbers n = concat(x,y) such that the product x*y | n. Leading zeros in y allowed. 5

%I

%S 11,12,15,24,36,101,102,104,105,110,120,125,150,208,240,306,315,360,

%T 735,1001,1002,1004,1005,1008,1010,1020,1025,1040,1050,1100,1125,1200,

%U 1250,1352,1500,1734,2016,2080,2400,3006,3015,3024,3060,3150,3375,3600,6048,7007

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

%C 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.

%C Other examples:

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

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

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

%H Paolo P. Lava and Giovanni Resta, <a href="/A255725/b255725.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Paolo P. Lava)

%e 15 = concat(1,5); 1*5 = 5 and 15 / 5 = 3.

%e 36 = concat(3,6); 3*6 = 18 and 36 / 18 = 2.

%e 9072 = concat(9,072); 9*72 = 648 and 9072 / 648 = 14.

%p with(numtheory); P:=proc(q) local a,b,i,n;

%p for n from 1 to q do for i from 1 to ilog10(n) do

%p a:=trunc(n/10^i); b:=n-a*10^i;

%p if a*b>0 then if type(n/(a*b),integer) then print(n);

%p fi; fi; od; od; end: P(10^9);

%t 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 *)

%Y Cf. A007602, A255726, A256518.

%K nonn,base

%O 1,1

%A _Paolo P. Lava_, Apr 01 2015

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Last modified July 24 05:07 EDT 2019. Contains 325290 sequences. (Running on oeis4.)