A257778 -id:A257778 - cp's OEIS frontend

A106039 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

Offset: 1

Eric Angelini, Jun 07 2005

Given an integer -1 < k < 10, n is a Belgian-k number if an infinite sequence in ascending order can be constructed starting at k and including n, and the first differences of that sequence give the base 10 digits of n repeatedly and no others.
Mauro Fiorentini (see Angelini link) explains that all base 10 Harshad numbers (A005349) are also Belgian-0 numbers. - Alonso del Arte, Feb 13 2014
A257778(a(n)) = A257770(a(n),0) = 0. - Reinhard Zumkeller, May 08 2015
Every integer in this sequence is also a Belgian-k number, where k is the sum of digits of the integer. - Davide Rotondo, Jun 12 2024

			13 is a Belgian-0 number because of the sequence
0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, ...
the first differences of which are
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
176 is a Belgian-0 number because, starting from 0 (the seed), 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.... (sequence)
.1.7.6..1..7..6..1..7..6..1..7..........7...6...1...7.. (first differences)
14 is not a Belgian number because, although we can construct a sequence with the required starting point and the required first differences (namely 0, 1, 5, 6, 10, 11, 15, ...), that sequence does not contain 14.

Vincenzo Librandi and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from Vincenzo Librandi
Eric Angelini, Belgian numbers.
Eric Angelini, Belgian Numbers [Cached copy with permission]

Cf. A257782 (complement), A253717 (primes).
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770, A257778.

a106039 n = a106039_list !! (n-1)
a106039_list = filter belge0 [0..] where
   belge0 n = n == (head $ dropWhile (< n) $
                    scanl (+) 0 $ cycle ((map (read . return) . show) n))
-- Reinhard Zumkeller, May 07 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 120, belgianQ[#, 0] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A257770 Table read by rows: numbers k, 0<=k<=9, such that n is a Belgian-k number.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 3, 5, 6, 8, 9, 0, 1, 4, 5, 8, 9, 3, 4, 8, 9, 2, 3, 8, 9, 1, 2, 8, 9, 0, 1, 8, 9, 0, 8, 9, 8, 9, 0, 2, 4, 6, 8, 0, 1, 3, 4, 6, 7, 9, 0

Offset: 0

Reinhard Zumkeller, May 08 2015

See A106039 for definition of Belgian-k numbers;
T(n,0) = A257778(n); T(n,A257773(n)) = A257779(n);
m is a Belgian-0 number iff T(m,0) = 0.

			The first 100 rows:
.   0 | 0                     25 | 2,4,9       50 | 0,5     75 | 3,8
.   1 | 0,1                   26 | 0,2,8       51 | 3,4,9   76 | 4
.   2 | 0,2                   27 | 0,7,9       52 | 3,5     77 | 0,7
.   3 | 0,3                   28 | 6,8         53 | 0,5,8   78 | 3
.   4 | 0,4                   29 | 5,7         54 | 0,4,9   79 | 8
.   5 | 0,5                   30 | 0,3,6,9     55 | 0,5     80 | 0,8
.   6 | 0,6                   31 | 0,3,4,7,8   56 | 1,7     81 | 0,1,9
.   7 | 0,7                   32 | 2,4,7,9     57 | 4,9     82 | 2,4
.   8 | 0,8                   33 | 0,3,6,9     58 | 1,6     83 | 6,9
.   9 | 0,9                   34 | 3,6         59 | 3       84 | 0,4
.  10 | 0,1,2,3,4,5,6,7,8,9   35 | 0,3,8       60 | 0,6     85 | 7
.  11 | 0,1,2,3,4,5,6,7,8,9   36 | 0,6,9       61 | 5,6     86 | 2,8
.  12 | 0,2,3,5,6,8,9         37 | 4,7         62 | 0,6,8   87 | 4
.  13 | 0,1,4,5,8,9           38 | 2,5         63 | 0,3,9   88 | 0,8
.  14 | 3,4,8,9               39 | 0,3         64 | 4,8     89 | 4
.  15 | 2,3,8,9               40 | 0,4,8       65 | 4       90 | 0,9
.  16 | 1,2,8,9               41 | 1,2,6,7     66 | 0,6     91 | 1,2
.  17 | 0,1,8,9               42 | 0,2,6,8     67 | 2,9     92 | 4,6
.  18 | 0,8,9                 43 | 1,4,8       68 | 6       93 | 0,9
.  19 | 8,9                   44 | 0,4,8       69 | 3,9     94 | 3,7
.  20 | 0,2,4,6,8             45 | 0,5,9       70 | 0,7     95 | 2
.  21 | 0,1,3,4,6,7,9         46 | 2,6         71 | 0,7,8   96 | 6
.  22 | 0,2,4,6,8             47 | 3           72 | 0,2,9   97 | 1,8
.  23 | 1,3,6,8               48 | 0,8         73 | 3,6     98 | 4
.  24 | 0,4,6                 49 | 6           74 | 1,8     99 | 0,9  .

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A257773 (row lengths), A257778 (min per row), A257778 (max per row),

Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.

a257770 n k = a257770_tabf !! n !! k
a257770_row n = filter belge [0..9] where
   belge k = n == (head $ dropWhile (< n) $
                  scanl (+) k $ cycle $ (map (read . return) . show) n)
a257770_tabf = map a257770_row [0..]

A257779 Largest k, such that n is a Belgian-k number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 8, 8, 6, 9, 8, 9, 8, 7, 9, 8, 9, 9, 6, 8, 9, 7, 5, 3, 8, 7, 8, 8, 8, 9, 6, 3, 8, 6, 5, 9, 5, 8, 9, 5, 7, 9, 6, 3, 6, 6, 8, 9, 8, 4, 6, 9, 6, 9, 7, 8, 9, 6, 8, 8, 4, 7, 3, 8, 8, 9, 4, 9, 4, 7

Offset: 0

Reinhard Zumkeller, May 08 2015

nonn

See A106039 for definition of Belgian-k numbers;
a(n) = A257770(n,A257773(n));
a(n) >= A257778(n); a(A257785(n)) = A257778(A257785(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A106039, A257770, A257773, A257778.

Haskell
```
a257779 = last . a257770_row
```

A257785 Numbers that are Belgian-k for exactly one k.

Original entry on oeis.org

0, 47, 49, 59, 65, 68, 76, 78, 79, 85, 87, 89, 95, 96, 98, 167, 177, 187, 193, 194, 239, 267, 268, 269, 286, 287, 293, 298, 299, 338, 349, 359, 367, 379, 394, 397, 398, 418, 437, 438, 458, 478, 479, 492, 497, 498, 499, 507, 528, 529, 536, 547, 548, 560, 568

Offset: 1

Reinhard Zumkeller, May 08 2015

nonn

See A106039 for definition of Belgian-k numbers;
A257773(a(n)) = 1;
A257778(a(n)) = A257779(a(n)).

			Let B(n) = A257778(a(n)), the singleton of a(n)-th row in A257770:
.   n |  a(n) | B(n)        n |  a(n) | B(n)         n |  a(n) | B(n)
.  ---+-------+-----     -----+-------+-----     ------+-------+-----
.   1 |     0 |   0       100 |   763 |   4       1000 |  6702 |   6
.   2 |    47 |   3       101 |   766 |   6       1001 |  6706 |   5
.   3 |    49 |   6       102 |   768 |   5       1002 |  6709 |   8
.   4 |    59 |   3       103 |   769 |   8       1003 |  6719 |   3
.   5 |    65 |   4       104 |   779 |   6       1004 |  6725 |   5
.   6 |    68 |   6       105 |   781 |   6       1005 |  6728 |   6
.   7 |    76 |   4       106 |   785 |   5       1006 |  6730 |   4
.   8 |    78 |   3       107 |   787 |   2       1007 |  6736 |   4
.   9 |    79 |   8       108 |   788 |   6       1008 |  6742 |   3
.  10 |    85 |   7       109 |   789 |   6       1009 |  6747 |   3
.  11 |    87 |   4       110 |   790 |   6       1010 |  6748 |   6
.  12 |    89 |   4       111 |   793 |   7       1011 |  6752 |   6
.  13 |    95 |   2       112 |   794 |   7       1012 |  6753 |   6
.  14 |    96 |   6       113 |   795 |   2       1013 |  6755 |   3
.  15 |    98 |   4       114 |   796 |   4       1014 |  6756 |   6
.  16 |   167 |   6       115 |   797 |   8       1015 |  6758 |   6
.  17 |   177 |   4       116 |   798 |   6       1016 |  6766 |   3
.  18 |   187 |   2       117 |   799 |   8       1017 |  6768 |   5
.  19 |   193 |   1       118 |   805 |   4       1018 |  6770 |   4
.  20 |   194 |   2       119 |   807 |   4       1019 |  6772 |   5  .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257770, A257773, A007088, A257778, A257779.

a257785 n = a257785_list !! (n-1)
a257785_list = filter ((== 1) . a257773) [0..]

A257782 Numbers that are not Belgian-0 numbers.

Original entry on oeis.org

14, 15, 16, 19, 23, 25, 28, 29, 32, 34, 37, 38, 41, 43, 46, 47, 49, 51, 52, 56, 57, 58, 59, 61, 64, 65, 67, 68, 69, 73, 74, 75, 76, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92, 94, 95, 96, 97, 98, 103, 104, 105, 107, 109, 113, 115, 116, 118, 119, 122, 124, 125

Offset: 1

Reinhard Zumkeller, May 08 2015

A257778(a(n)) = A257770(a(n),0) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257770, A257778, A106039 (complement).

a257782 n = a257782_list !! (n-1)
a257782_list = filter ((> 0) . a257778) [0..]

cp's OEIS Frontend

A106039 Belgian-0 numbers.

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A257770 Table read by rows: numbers k, 0<=k<=9, such that n is a Belgian-k number.

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A257779 Largest k, such that n is a Belgian-k number.

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A257785 Numbers that are Belgian-k for exactly one k.

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A257782 Numbers that are not Belgian-0 numbers.

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