A255725 -id:A255725 - cp's OEIS frontend

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

Paolo P. Lava, Apr 01 2015

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)?

			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.

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

Cf. A007602, A255725, A256518.

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);

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 *)

A256518 Consider numbers n = concat(x,y,z) such that the product xyz | n. Leading zeros in y and z allowed. Sequence lists numbers that admit different concatenations.

Original entry on oeis.org

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

Paolo P. Lava, Apr 01 2015

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.

			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.

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

Cf. A007602, A255725, A255726.

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);

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 *)

A257897 Numbers n such that n = concat(a,b) and n | a^b + b^a , with a>0 and b>0.

Original entry on oeis.org

63, 103, 128, 147, 155, 212, 272, 292, 351, 452, 486, 497, 525, 527, 537, 584, 607, 624, 648, 729, 979, 999, 1024, 1296, 1323, 1359, 1533, 1541, 1575, 1809, 1872, 2048, 2050, 2107, 2187, 2448, 2512, 2537, 2564, 2763, 2793, 2886, 3072, 3357, 3927, 4096, 4263, 4284

Offset: 1

Paolo P. Lava, May 12 2015

We can have different solutions for the same number. E.g.: 2048 divides both (20^48 + 48^20) and (204^8 + 8^204). The same occurs for 4096, 4263, 16807, 32768, 96957, 156672, 186624, 252081, 262144, 270729, 352947, 390624 … The first number with 3 different concatenations is 186624 that divides (18^6624 + 6624^18), (186^624 + 624^186) and (1866^24 + 24^1866).

			6^3 + 3^6 = 945 and 945 / 63 = 15;
10^3 + 3^10 = 60049 and 60049 / 103 = 583;
12^8 + 8^12 = 69149458432 and 69149458432 / 128 = 540230144; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..400

Cf. A253824, A253825, A255725, A255726.

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>0 and b>0 then if type((a^b+b^a)/n,integer)
then print(n); break; fi; fi; od; od; end: P(10^9);

tst[n_]:=Catch@Block[{a,b}, Do[a=Floor[n/10^k]; b=Mod[n,10^k]; If[Mod[ PowerMod[a, b, n] + PowerMod[b, a, n], n]==0, Throw@True], {k, IntegerLength[n]-1}]; False]; Select[Range@1000, tst] (* Giovanni Resta, May 12 2015 *)

A258319 Numbers n such that n = concat(a,b) and n = phi(n) + phi(a) + phi(b), with a>0 and b>0, where phi(n) is the Euler totient function of n.

Original entry on oeis.org

25, 177, 1177, 2501, 17105, 21337, 22681, 32581, 217009, 409501, 561601, 577501, 861841, 1025821, 1401841, 1738081, 2836465, 8331361, 10284193, 19971901, 20103001, 27835921, 31949921, 34897501, 100763053, 107314217, 111512701, 121806001, 150658561, 155874001

Offset: 1

Paolo P. Lava, May 26 2015

			25 = concat(2,5); phi(25) + phi(2) + phi(5) = 20 + 1 + 4 = 25;
177 = concat(1,77); phi(177) + phi(1) + phi(77) = 116 + 1 + 60 = 177; etc.

Giovanni Resta, Table of n, a(n) for n = 1..50

Cf. A000010, A239562, A249766, A251605, A251860, A253824, A253825, A255725, A255726, A257897.

with(numtheory); P:=proc(q) local a, b, i; global 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>0 and b>0 then if phi(n)+phi(a)+phi(b)=n
then print(n); break; fi; fi; od; od; end: P(10^9);

a(18) inserted by Giovanni Resta, May 27 2015

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

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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.

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A257897 Numbers n such that n = concat(a,b) and n | a^b + b^a , with a>0 and b>0.

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A258319 Numbers n such that n = concat(a,b) and n = phi(n) + phi(a) + phi(b), with a>0 and b>0, where phi(n) is the Euler totient function of n.

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A256518 Consider numbers n = concat(x,y,z) such that the product xyz | n. Leading zeros in y and z allowed. Sequence lists numbers that admit different concatenations.