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.
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
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
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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);
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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