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A265740 a(1)=1; a(n+1) is the smallest positive integer not yet used such that all the digits of a(n) and a(n+1) are present in the decimal expansion (including any leading and trailing zeros) of a(n)/a(n+1).

%I A265740 #29 Dec 23 2015 14:13:35
%S A265740 1,6,13,10,14,17,7,8,19,23,21,29,34,31,3,38,28,46,47,35,39,49,43,51,
%T A265740 42,41,48,53,26,12,57,58,59,2,61,24,68,11,52,63,22,69,62,71,56,65,76,
%U A265740 81,44,67,64,83,85,78,77,79,72,70,80,87,84,86,89,9,91,92,73
%N A265740 a(1)=1; a(n+1) is the smallest positive integer not yet used such that all the digits of a(n) and a(n+1) are present in the decimal expansion (including any leading and trailing zeros) of a(n)/a(n+1).
%C A265740 Conjecture: a(n) is a permutation of the natural numbers.
%C A265740 The following table shows:
%C A265740 C = number of terms calculated
%C A265740 F = first term that is missing
%C A265740 C           F      F/C
%C A265740 1000        5    0.005
%C A265740 2000       50    0.025
%C A265740 5000     1650    0.330
%C A265740 10000    1650    0.165
%C A265740 20000    2475    0.124
%C A265740 50000   24750    0.495
%C A265740 100000 100000    1.000
%C A265740 200000 199800    0.999
%C A265740 500000 499500    0.999
%C A265740 which seems to support the conjecture.
%H A265740 Lars Blomberg, <a href="/A265740/b265740.txt">Table of n, a(n) for n = 1..10000</a>
%e A265740    1/6  = 0.1666... (1 and 6 are visible on the right-hand side)
%e A265740    6/13 = 0.461538461538... (6, 1 and 3 are visible)
%e A265740   13/10 = 1.30 (trailing zeros are included)
%e A265740   10/14 = 0.7142857142... (1, 0 and 4)
%e A265740   14/17 = 0.8235294117... (1, 4 and 7)
%e A265740   17/7  = 2.4285714285... (1 and 7)
%e A265740    7/8  = 0.875 (7 and 8)
%e A265740    ...
%t A265740 f[n_] := Block[{a = {1}, k}, Do[k = If[MissingQ@ #, Max@ a, #] &@ SelectFirst[Range@ Max@ a, ! MemberQ[a, #] &]; While[Or[! AllTrue[Join[IntegerDigits@ a[[i - 1]], IntegerDigits@ k], MemberQ[Union@ Flatten@ Prepend[First@ #, If[Last@ # <= 0, 0, Nothing]] &@ If[Depth@ First@ # < 3, Insert[#, 0, {1, 1}], #] &@ RealDigits[a[[i - 1]]/k], #] &], MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 67 (* Version 10.2 *)
%t A265740 f[n_] := Block[{a = {1}, k}, Do[k = 1; While[Or[If[# == 1, False, True] &[Times @@ Boole[MemberQ[Union@ Flatten@ Prepend[First@ #, If[Last@ # <= 0, 0]] &@ If[Depth@ First@ # < 3, Insert[#, 0, {1, 1}], #] &@ RealDigits[a[[i - 1]]/k], #] & /@ Join[IntegerDigits@ a[[i - 1]], IntegerDigits@ k]]], MemberQ[a, k]], k++]; AppendTo[a, k], {i, 2, n}]; a]; f@ 67 (* _Michael De Vlieger_, Dec 16 2015, Version 6 *)
%Y A265740 See A265756 for another version.
%Y A265740 See also A257664.
%K A265740 nonn,base
%O A265740 1,2
%A A265740 _Eric Angelini_, submitted by _Lars Blomberg_, Dec 15 2015
%E A265740 Corrected values for n>=58 by _Lars Blomberg_, Dec 16 2015