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A257172
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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.
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0
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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