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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. 0

%I #8 May 01 2015 13:14:55

%S 11424,13248,14112,16128,16632,17136,18144,41328,91728,101112,102144,

%T 102816,104832,106272,111012,111375,112288,112896,114048,114240,

%U 114912,116160,116928,123120,132480,140112,141120,161280,166320,171171,171360,181440,203112,204288,204336,220416,231012,233772,239616

%N 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.

%e 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.

%p with(numtheory); P:=proc(q) local a,ab,b,c,cd,d,i,j,k,m,n,v,w,z;

%p v:=array(1..10, 1..4); w:=[]; for n from 1 to q do j:=0;

%p for i from 1 to ilog10(n) do c:=(n mod 10^i); ab:=trunc(n/10^i);

%p for k from 1 to ilog10(ab) do d:=(ab mod 10^k); cd:=trunc(ab/10^k);

%p for z from 1 to ilog10(cd) do a:=trunc(cd/10^z); b:=cd-a*10^z;

%p if a*b*c*d>0 then if type(n/(a*b*c*d), integer) then j:=j+1;

%p w:=sort([a,b,c,d]); for m from 1 to 4 do v[j,m]:=w[m]; od;

%p 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]

%p then j:=j-1; break; fi; od; fi; fi; od; od; od;

%p if j>1 then print(n); fi; od; end: P(10^9);

%t 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

%Y Cf. A256518.

%K nonn,base

%O 1,1

%A _Paolo P. Lava_ and _Robert G. Wilson v_, Apr 17 2015

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)