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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).
3
1, 6, 13, 10, 14, 17, 7, 8, 19, 23, 21, 29, 34, 31, 3, 38, 28, 46, 47, 35, 39, 49, 43, 51, 42, 41, 48, 53, 26, 12, 57, 58, 59, 2, 61, 24, 68, 11, 52, 63, 22, 69, 62, 71, 56, 65, 76, 81, 44, 67, 64, 83, 85, 78, 77, 79, 72, 70, 80, 87, 84, 86, 89, 9, 91, 92, 73
OFFSET
1,2
COMMENTS
Conjecture: a(n) is a permutation of the natural numbers.
The following table shows:
C = number of terms calculated
F = first term that is missing
C F F/C
1000 5 0.005
2000 50 0.025
5000 1650 0.330
10000 1650 0.165
20000 2475 0.124
50000 24750 0.495
100000 100000 1.000
200000 199800 0.999
500000 499500 0.999
which seems to support the conjecture.
LINKS
EXAMPLE
1/6 = 0.1666... (1 and 6 are visible on the right-hand side)
6/13 = 0.461538461538... (6, 1 and 3 are visible)
13/10 = 1.30 (trailing zeros are included)
10/14 = 0.7142857142... (1, 0 and 4)
14/17 = 0.8235294117... (1, 4 and 7)
17/7 = 2.4285714285... (1 and 7)
7/8 = 0.875 (7 and 8)
...
MATHEMATICA
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 *)
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 *)
CROSSREFS
See A265756 for another version.
See also A257664.
Sequence in context: A070396 A359091 A130012 * A090324 A106623 A115010
KEYWORD
nonn,base
AUTHOR
Eric Angelini, submitted by Lars Blomberg, Dec 15 2015
EXTENSIONS
Corrected values for n>=58 by Lars Blomberg, Dec 16 2015
STATUS
approved