A106792 -id:A106792 - 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..]

A106596 Belgian-3 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a106596 n = a106596_list !! (n-1)
a106596_list = filter belge3 [3..] where
   belge3 x = x == (head $ dropWhile (< x) $
                    scanl (+) 3 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 200, belgianQ[#, 3] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A106439 Belgian-1 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a106439 n = a106439_list !! (n-1)
a106439_list = filter belge1 [1..] where
   belge1 x = x == (head $ dropWhile (< x) $
                    scanl (+) 1 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 202, belgianQ[#, 1] &] (* Robert G. Wilson v, May 06 2011 *)

A106518 Belgian-2 numbers.

Original entry on oeis.org

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, 141, 142, 143

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a106518 n = a106518_list !! (n-1)
a106518_list = filter belge2 [2..] where
   belge2 x = x == (head $ dropWhile (< x) $
                    scanl (+) 2 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 145, belgianQ[#, 2] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A106631 Belgian-4 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a106631 n = a106631_list !! (n-1)
a106631_list = filter belge4 [4..] where
   belge4 x = x == (head $ dropWhile (< x) $
                    scanl (+) 4 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 199, belgianQ[#, 4] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A107014 Belgian-6 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a107014 n = a107014_list !! (n-1)
a107014_list = filter belge6 [6..] where
   belge6 x = x == (head $ dropWhile (< x) $
                    scanl (+) 6 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 159, belgianQ[#, 6] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A107018 Belgian-7 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a107018 n = a107018_list !! (n-1)
a107018_list = filter belge7 [7..] where
   belge7 x = x == (head $ dropWhile (< x) $
                     scanl (+) 7 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 206, belgianQ[#, 7] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A107032 Belgian-8 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a107032 n = a107032_list !! (n-1)
a107032_list = filter belge8 [8..] where
   belge8 x = x == (head $ dropWhile (< x) $
                    scanl (+) 8 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 148, belgianQ[#, 8] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

A107043 Belgian-9 numbers.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Jun 07 2005

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

See A106039 for definition and link.
Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770.

a107043 n = a107043_list !! (n-1)
a107043_list = filter belge9 [9..] where
   belge9 x = x == (head $ dropWhile (< x) $
                    scanl (+) 9 $ cycle (map (read . return) $ show x))
-- Reinhard Zumkeller, May 08 2015

belgianQ[n_, k_] := If[n < k, False, Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - k, Plus @@ id]] ]]; Select[ Range@ 155, belgianQ[#, 9] &] (* Robert G. Wilson v, May 06 2011 *)

Offset changed by Reinhard Zumkeller, May 08 2015

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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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A106596 Belgian-3 numbers.

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A106439 Belgian-1 numbers.

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A106518 Belgian-2 numbers.

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A106631 Belgian-4 numbers.

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A107014 Belgian-6 numbers.

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A107018 Belgian-7 numbers.

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