

A106039


Belgian0 numbers.


19



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
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OFFSET

1,3


COMMENTS

Given an integer 1 < k < 10, n is a Belgiank 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 Belgian0 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 Belgian0 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 Belgian0 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 !! (n1)
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).
Belgiank numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
Cf. A257770, A257778.
Sequence in context: A053432 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



