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

1,3

Comments

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

Links

Vincenzo Librandi and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from Vincenzo Librandi

E. Angelini, Belgian numbers.

E. Angelini, Belgian Numbers [Cached copy with permission]

Example

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.

Mathematica

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 *)

Prog

(Haskell)
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

Crossrefs

Cf. A257782 (complement), A253717 (primes).

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

Cf. A257770, A257778.

Sequence in context: A349999 A261888 A154125 * A151767 A213517 A173899

Adjacent sequences: A106036 A106037 A106038 * A106040 A106041 A106042

Keyword

base,easy,nonn

Author

Eric Angelini, Jun 07 2005

Extensions

Offset changed by Reinhard Zumkeller, May 08 2015

Status

approved