cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Apr 01 2015

Keywords

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

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 *)

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

Original entry on oeis.org

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

Views

Author

Paolo P. Lava, Apr 01 2015

Keywords

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

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 *)

A257172 Consider numbers n = concat(w,x,y,z) such that w*x*y*z | n. Leading zeros in x, y and z allowed. Sequence lists numbers that admit at least two such concatenations.

Original entry on oeis.org

11424, 13248, 14112, 16128, 16632, 17136, 18144, 41328, 91728, 101112, 102144, 102816, 104832, 106272, 111012, 111375, 112288, 112896, 114048, 114240, 114912, 116160, 116928, 123120, 132480, 140112, 141120, 161280, 166320, 171171, 171360, 181440, 203112, 204288, 204336, 220416, 231012, 233772, 239616
Offset: 1

Views

Author

Paolo P. Lava and Robert G. Wilson v, Apr 17 2015

Keywords

Examples

			11424 / (1*1*4*24)=119, 11424 / (1*1*42*4)=68 and 11424 / (1 14*2*4)  but 11424 / (11*4*2*4) is 357/11, not an integer. So 11424 is the concatenation of three sets of four integers whose products divide 11424.

Crossrefs

Cf. A256518.

Programs

  • Maple
    with(numtheory); P:=proc(q) local a,ab,b,c,cd,d,i,j,k,m,n,v,w,z;
v:=array(1..10, 1..4); 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 d:=(ab mod 10^k); cd:=trunc(ab/10^k);
for z from 1 to ilog10(cd) do a:=trunc(cd/10^z); b:=cd-a*10^z;
if a*b*c*d>0 then if type(n/(a*b*c*d), integer) then j:=j+1;
w:=sort([a,b,c,d]); for m from 1 to 4 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] and v[m,4]=v[j,4]
then j:=j-1; break; fi; od; fi; fi; od; 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, k}], FromDigits@ Take[id, {k + 1, lng}]}, {i, 1, lng - 3}, {j, i + 1, lng - 2}, {k, j + 1, lng - 1}], 4]]; Count[IntegerQ /@ (n/t), True] > 1]; k = 1000; lst = {}; While[k < 100000001, If[fQ@ k, AppendTo[lst, k]]; k++]; lst
Showing 1-3 of 3 results.

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