Belgian Numbers (formerly Eric Numbers).
176 is an “Belgian-0
number” because, starting from 0, one can build a sequence containing
176 in this way:
0 1 8 14 15 22 28 29
36 42 43 50 ... 155 162 168 169 176 ...
1 7 6 1
7 6 1
7 6 1
7 ... 7
6 1 7
The “first
differences” building rule is easy to understand. The above example
shows that one doesn’t have to add the full digit-pattern [1+7+6] to produce the according Belgian number: 176
already appears when 7 is added to the previous integer – not after 6 is
added.
Here are the first Belgian-0
numbers:
0 1 2 3 4 5
6 7 8 9 10 11 12 13 17 18 20 21 22 24 26 27 30 31 33 35 36 39 40 42 44 45 48 50
53 54 55 60 62 63 66 70 71 72 77 80 81 84 88 90 93 99 100 101 102 106 108 110
111 112 114 117 120 ...
Here is another
example in order to explain how the above sequence works. Take it’s integer 17 for instance; 17 is an Belgian-0
number because 17 belongs to this infinite sequence:
0 1 8 9 16 17 24 25 32 ...
1 7 1 7 1
7 1 7
_______
Now, we have started from
0 (zero) but we could have started from any other “seed”, ranging from 0 to 9 (in Belgian number’s world, seeds cannot
be greater than 9 – this will be explained later).
Belgian-1 numbers (seed in bold):
1 10 11 13
16 17 21 23 41 43 56 58 74 81 91 97 100 101 106 110 111 113 115 121 122 130 131
137 142 155 157 161 170 171 172 178 179 181 184 188 193 201 ...
179, for instance, is an Belgian-1
number because (seed in bold):
1 2 9 18 19
26 35 36 43 52 53 ... 155 162 171 172 179.
1 7 9 1
7 9 1 7 9
1 ... 7
9 1 7
Belgian-2 numbers:
2 10 11 12
15 16 20 22 25 26 32 38 41 42 46 67 72 82 86 91 95 100 101 102 103 105 107 110
111 112 113 115 116 120 121 122 123 124 125 130 131 132 134 136 138 142 143 ...
138, for instance, is an
Belgian-2 number (seed
in bold):
2 3 6 14 15
18 26 27 30 38 39 ... 122 123 126 134 135 138.
1 3 8 1
3 8 1
3 8 1
... 1 3
8 1 3
Belgian-3 numbers:
3 10 11 12
14 15 21 23 30 31 33 34 35 39 47 51 52 59 63 69 73 75 78 94 100 101 102 103 104
105 107 110 111 112 113 115 116 120 123 133 141 146 147 151 153 154 158 159 163 164 166 168 183
185 191 196 ...
159, for instance, is an
Belgian-3 number (seed
in bold):
3 4 9 18 19 24 33 34 39 48 49 ... 139 144 153 154 159.
1 5 9 1
5 9 1
5 9 1 ... 5
9 1 5
Belgian-4 numbers:
4 10 11 13
14 20 21 22 24 25 31 32 37 40 43 44 51 54 57 64 65 76 82 84 87 89 92 98 100 101
104 110 111 112 114 116 121 122 124 125 127 128 137 140 141 142 144 145 148 149 151 154 158 172 177
191 196 ...
149, for instance, is an
Belgian-4 number (seed
in bold):
4 5 9 18 19 23 32 33 37 46 47 ... 131 135 144 145 149.
1 4 9 1
4 9 1
4 9 1
... 4 9
1 4
Belgian-5 numbers:
5 10 11 12
13 29 38 45 50 52 53 55 61 100 101 102 110 111 114 120 121 124 125 130 131 132
134 135 136 137 138 139 140 145 148 150 151 160 174 175 182 186 191 195 211
...
148, for instance, is an
Belgian-5 number (seed
in bold):
5 6 10 18 19 23 31 32 36 44 45 ... 127 135 136 140 148.
1 4 8
1 4 8
1 4 8
1 ... 8
1 4 8
Belgian-6 numbers:
6 10 11 12
20 21 22 23 24 28 30 33 34 36 41 42 46 49 58 60 61 62 66 68 73 83 92 96 100 101
102 103 110 111 112 113 114 118 120 121 122 123 126 127 128 129 130 131 132 133
134 136 138 143 150
155 156 ...
138, for instance, is an
Belgian-6 number (seed
in bold):
6 7 10 18 19 22 30 31 34 42 43 ... 118 126 127 130 138.
1 3 8
1 3 8
1 3 8
1 ... 8
1 3 8
Belgian-7 numbers:
7 10 11 21 27
29 31 32 37 41 56 70 71 77 85 94 100 101 103 106 110 111 112 113 117 118 119
122 127 128 131 133
143 152 173 176 201 205 ...
128, for instance, is an
Belgian-7 number (seed
in bold):
7 8 10 18 19 21 29 30 32 40 41 ... 109 117 118
120 128.
1 2 8
1 2 8
1 2 8
1 ... 8
1 2 8
Belgian-8 numbers:
8 10 11 12
13 14 15 16 17 18 19 20 22 23 26 28 31 35 40 42 43 44 48 53 62 64 71 74 75 79
80 86 88 97 100 101 102 104 105 106 108 109 110 111 112 113 115 117 118 119 120
121 123 126 129 132 135 139 141 142 144 149 152 153 154 157 159 161 ...
119, for instance, is an
Belgian-8 number (seed
in bold):
8 9 10 19 20 21 30 31 32 41 42 ... 107 108 109 118 119.
1 1 9 1 1 9 1 1 9 1
... 1 1 9 1
Belgian-9 numbers:
9 10 11 12
13 14 15 16 17 18 19 21 25 27 30 32 33 36 45 51 54 57 63 67 69 72 81 83 90 93
99 100 101 102 104 105 108 109 110 111 115 117 119 120 121 122 123 124 126 129
130 135 139 140 141 142 144 146 149 153 159 161 162 164 165 166 169 ...
149, for instance, is an
Belgian-9 number (seed
in bold):
9 10 14 23 24 28 37 38 42 51 52 ... 126 135 136 140 149.
1 4
9 1 4
9 1 4
9 1 ...
9 1 4 9
_______
Two types of Self-Belgian
Numbers (SBN) could be
also defined – if you are not asleep yet.
The first type (SBN-1) would only consist in Belgian numbers whose
building sequence begins with the same seed as their leftmost digit.
179 is an example of Self-Belgian Number of
type-1. The “seed” is 1 because 1 is the leftmost digit of 179.
Here is the complete sequence leading to 179:
_
1 2 9 18 19 26 35 36 43 52 53 60 69 70 77 86 87 94 103
104
1 7 9
1 7 9
1 7 9
1 7 9
1 7 9
1 7 9
1 7
___
111 120
121 128 137 138 145 154 155 162 171 172 179.
9
1 7 9
1 7 9
1 7 9
1 7
And here are the
first Self-Belgian Numbers of type-1 (SBN-1):
0 1 2 3 4 5 6 7 8 9
10 11 13 16 17 20 22 25 26 30 31 33 34 35 39 40 43 44 50 52 53 55 60 61 62 66 68 70 71 77 80 86 88 90
93 99 100 101 106 110 111 113 115 121 122 130 131 137 142 155 157 161 170 171
172 178 179 181 184 188 193 201 ...
Again, this sequence
should be red like this: 68 (for instance) is an Belgian‑6
number; 70 is an Belgian‑7 number; (and so are also 71 and 77); 80 is an Belgian‑8
number, etc. All the above SBN-1 integers use their leftmost digit as seed
for their building sequence.
_______
The second types of Self Belgian Numbers (my favorite,
SBN-2) are numbers who fully show all their digits (in the same order)
at the beginning of their building sequence – and not only their
leftmost one. 61 is the first such
integer:
___ __
6
12 13 19 20 26 27 33 34 40 41 47 48 54 55 61.
6 1 6
1 6 1
6 1 6
1 6 1
6 1 6
As one can see, the
seed remains 6 – and not 61. If we allow seeds to have more than
one digit, then all integers would be SEN-2, right from the beginning of their
building sequence! This is why seeds cannot be greater than 9.
The beginning of the
SBN-2 sequence looks like this:
61 71 918 3612 5101 8161 ...
Again, this last
integer belongs to the SBN-2 family because its building sequence shows at the
very beginning all it’s digits (in the same order):
______ ____
8 16 17 23 24 32
33 39 40 ... 8145 8151 8152 8160 8161
8 1 6 1 8
1 6 1
... 6 1
8 1
_______
More terms (remarks and corrections) are welcome (here).
None of those
sequences are yet in the OEIS. They will be submitted soon. (They are now)
The concept of Belgian
numbers came to the lousy author after his discovery of the Keith
Numbers (or Repfigits), there.
_______
A comment
on this from Eugene McDonnell, here.
And a
wonderful page from Jean-Paul Davalan, with lots of applets, there.
___________
[copyleft:
E. Angelini, Brussels, Belgium – June 7t, 2005.]
Back to the main page (in
French)