A257770 -id:A257770 - cp's OEIS frontend

A107043 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

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

A257773 Number of numbers k, 0 <= k <= 9, such that n is a Belgian-k number.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 7, 6, 4, 4, 4, 4, 3, 2, 5, 7, 5, 4, 3, 3, 3, 3, 2, 2, 4, 5, 4, 4, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 3, 3, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 1, 2, 2, 3, 3, 2, 2, 2, 1, 2, 1, 1, 2, 3, 2, 2, 2, 1

Offset: 0

Reinhard Zumkeller, May 08 2015

nonn

 See A106039 for definition of Belgian-k numbers.
row lengths in A257770;
a(A257785(n)) = 1;
For n > 1: a(A007088(n)) = 10.

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

Cf. A106039, A257770, A257785, A007088.

Haskell
```
a257773 = length . a257770_row
```

A257778 Smallest k, such that n is a Belgian-k number.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, May 08 2015

nonn

See A106039 for definition of Belgian-k numbers;
a(n) = A257770(n,0);
a(n) <= A257779(n); a(A257785(n)) = A257779(A257785(n));
conjecture: a(n) < 9;
a(A106039(n)) = 0; a(A257782(n)) > 0.

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

Cf. A106039, A257770, A257773, A257779, A257782.

Haskell
```
a257778 = head . a257770_row
```

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

A107043 Belgian-9 numbers.

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A257773 Number of numbers k, 0 <= k <= 9, such that n is a Belgian-k number.

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A257778 Smallest k, 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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