A106039 -id:A106039 - cp's OEIS frontend

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

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..]

A253717 Primes equal to their partial cyclical digital sum numbers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 53, 71, 101, 131, 157, 173, 181, 197, 211, 283, 431, 439, 457, 461, 487, 509, 571, 601, 643, 727, 911, 929, 1021, 1031, 1033, 1051, 1093, 1151, 1163, 1171, 1201, 1231, 1249, 1259, 1301, 1303, 1327, 1373, 1399, 1429, 1451, 1453, 1493

Offset: 1

V.J. Pohjola, May 02 2015

Subsequence of primes of A106039. - Michel Marcus, May 03 2015

			Prime(37) = 157 = (1+5+7)*12 + 1.
Prime(40) = 173 = (1+7+3)*15 + 1+7.
Prime(42) = 181 = (1+8+1)*18 + 1.

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

Cf. A257275.

Cf. A106039.

a253717 n = a253717_list !! (n-1)
a253717_list = filter ((== 1) . a010051') a106039_list
-- Reinhard Zumkeller, May 07 2015

terms = {}; (Do[p = Prime[n]; iD = IntegerDigits[p]; iD[[0]] = 0;
  a = Apply[Plus, iD]; pf = p - Mod[p, Floor[p/a]*a];
  (Do[pf = pf + Apply[Plus, iD[[i]]];
    If[pf == p, AppendTo[terms, pf]], {i, 0, IntegerLength[Prime[n]]}]), {n,
   1, 1000}]); Union[terms]

isok(n) = {my(v = divrem(n, sumdigits(n))[2]); if (!v, return (1)); d = digits(n); for (i=1, #d, v -= d[i]; if (!v, return (1));); return (0);}
lista(nn) = forprime (n=1, nn, if (isok(n), print1(n, ", "))); \\ Michel Marcus, May 03 2015

A107062 Union of set of Belgian-k numbers for k = 0..9 which begin with k.

Original entry on oeis.org

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

Offset: 0

Eric Angelini, Jun 07 2005

See A106039 for definition and link.

Cf. A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043, A107070.

belgianKQ[n_] := Block[{id = Join[{0}, IntegerDigits@ n]}, MemberQ[ Accumulate@ id, Mod[n - id[[2]], Plus @@ id]]]; Select[ Range@ 178, belgianKQ] (* Robert G. Wilson v, May 06 2011 *)

A107070 Numbers m with the following property. Suppose m = d1 d2 ... dk in base 10. Construct the sequence with first term d1 and successive differences d1 d2 ... dk d1 d2 ... dk d1 d2 ...; then this sequence has as its initial k digits d1 d2 ... dk and also contains the number m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 61, 71, 918, 3612, 5101, 8161, 12481, 51011, 248161, 361213, 5101111, 7141519, 8161723, 481617232, 2481617232, 4816172324, 5101111121, 24816172324, 51011111213, 71415192025, 612131516192, 816172324313, 3612131516192, 5101111121314, 6121315161920, 9181927283739

Offset: 1

Eric Angelini, Jun 07 2005

These are sometimes called Eric numbers or Belgian numbers. - N. J. A. Sloane, May 06 2011
Each digit {1..9} will produce a quasi-automorphic sequence. Thus this sequence is infinite. - Robert G. Wilson v, May 06 2011
The existence of the nine templates upon which the quasi-automorphic sequences are decided guarantees that no more than nine solutions exist for a given digit-length. The equidistribution of the ten base-ten digits within these templates predicts a long-term average of two solutions per digit-length. All nine solutions happen trivially for digit-length 1 (terms 1-9) and not again until digit-length 1899283 (terms 3594728-3594736). - Hans Havermann, May 27 2011, Aug 15 2011
The n-th term is prime for: n= 2, 3, 5, 7, 10, 11, 14, 15, 18, 19, 51, 55, 238, 907, 979, 1814, ..., . - Robert G. Wilson v, May 06 2011

			The following example shows why 61 is a member:
6.12.13.19.20.26.27.33.34.40.41.47.48.54.55.61... (sequence)
.6..1..6..1..6..1..6..1..6..1..6..1..6..1..6... (first differences)

E. Angelini, Belgian numbers.
E. Angelini, Belgian Numbers [Cached copy with permission]
J.-P. Davalan, Nombres belges [Includes applets to generate sequence]

Cf. A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043, A107062.

belgianDQ[n_] := Block[{id = IntegerDigits@ n, id1}, id1 = id[[1]]; MemberQ[ Accumulate@ Join[{0}, id], Mod[n - id1, Plus @@ id]] && id == Take[ Flatten[ IntegerDigits[ FoldList[#1 + #2 &, id1, id]]], Length@ id]] (* Robert G. Wilson v, May 06 2011 *)

Minor edits by N. J. A. Sloane, May 06 2011

A298984 Numbers k such that floor((10^p) / k) has digital sum k for some integer p.

Original entry on oeis.org

1, 3, 7, 8, 9, 13, 14, 22, 30, 33, 34, 43, 49, 51, 55, 56, 62, 66, 73, 76, 83, 90, 91, 92, 94, 95, 96, 98, 99, 103, 109, 113, 127, 129, 130, 132, 133, 137, 139, 141, 150, 154, 159, 169, 170, 174, 175, 177, 179, 180, 181, 185, 186, 192, 194, 202, 208, 211, 215

Offset: 1

Rémy Sigrist, Jan 31 2018

This sequence has similarities with A106039: here some partial sum of digits of 1/k equals k, there some partial (cyclical) sum of digits of k equals k.

A052268 is a subsequence.

			floor(1000 / 7) = 142 and 1 + 4 + 2 = 7, hence 7 belongs to this sequence.
floor(1 / 5) = 0 and floor ((10^p) / 5) = 2 for any p > 0, hence 5 does not belong to this sequence.

Cf. A007953, A052268, A106039.

is(n) = my (r=1/n, s=0); while (r, s+=floor(r); if (s==n, return (1), s>n, return (0)); r = frac(r)*10); return (0)

A357894 Integers k such that the sum of some number of initial decimal digits of sqrt(k) is equal to k.

Original entry on oeis.org

0, 1, 6, 10, 14, 18, 27, 33, 41, 43, 46, 55, 56, 62, 66, 69, 70, 77, 80, 87, 93, 98, 102, 108, 110, 123, 124, 145, 147, 149, 150, 154, 157, 162, 164, 165, 168, 176, 177, 179, 180, 182, 183, 197, 204, 213, 214, 219, 224, 236, 237, 242, 248, 251, 252, 261, 262, 263, 271, 274, 285, 295

Offset: 1

Gil Broussard, Oct 18 2022

For integers k that are squares of integers, "Sum of initial digits" includes digits to the left of the decimal point only, as there are no digits other than zero to the right of the decimal point. This constraint contributes terms 0 and 1 to the sequence.
For integers k with irrational sqrt(k), "Sum of initial digits" includes digits to the left of the decimal point and to the right of the decimal point.
"Initial digits" implies a sufficient number of digits to produce either a sum > k or a sum = k condition, halting at whichever condition occurs first (sum > k condition is discarded).

			41 is a term because sqrt(41) = 6.4031242374328... and 6+4+0+3+1+2+4+2+3+7+4+3+2 = 41.
42 is not a term because sqrt(42) = 6.480740698407860... and 6+4+8+0+7+4+0+6 = 35 and 6+4+8+0+7+4+0+6+9 = 44 (no sum of initial digits = 42).
144 is not a term because sqrt(144) = 12 (no digits to the right of the decimal), and 1+2 is not equal to 144.

Cf. A106039.

is(n) = { my (d=digits(sqrtint(n)), s=0); for (i=1, #d, s+=d[i]; if (s==n, return (1), s>n, return (0););); if (issquare(n), return (n==0);); my (n0=n); while (1, s+=sqrtint(n0*=100)%10; if (s==n, return (1), s>n, return (0););); } \\ Rémy Sigrist, Oct 19 2022

cp's OEIS Frontend

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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Haskell

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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A253717 Primes equal to their partial cyclical digital sum numbers.

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PARI

A107062 Union of set of Belgian-k numbers for k = 0..9 which begin with k.

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A107070 Numbers m with the following property. Suppose m = d1 d2 ... dk in base 10. Construct the sequence with first term d1 and successive differences d1 d2 ... dk d1 d2 ... dk d1 d2 ...; then this sequence has as its initial k digits d1 d2 ... dk and also contains the number m.

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Extensions

A298984 Numbers k such that floor((10^p) / k) has digital sum k for some integer p.

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