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A106039 Belgian-0 numbers.

%I A106039 #41 Jul 01 2024 22:21:30
%S A106039 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,
%T A106039 36,39,40,42,44,45,48,50,53,54,55,60,62,63,66,70,71,72,77,80,81,84,88,
%U A106039 90,93,99,100,101,102,106,108,110,111,112,114,117,120
%N A106039 Belgian-0 numbers.
%C A106039 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.
%C A106039 Mauro Fiorentini (see Angelini link) explains that all base 10 Harshad numbers (A005349) are also Belgian-0 numbers. - _Alonso del Arte_, Feb 13 2014
%C A106039 A257778(a(n)) = A257770(a(n),0) = 0. - _Reinhard Zumkeller_, May 08 2015
%C A106039 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
%H A106039 Vincenzo Librandi and Reinhard Zumkeller, <a href="/A106039/b106039.txt">Table of n, a(n) for n = 1..10000</a>, first 1000 terms from  Vincenzo Librandi
%H A106039 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/EricNumbers.htm">Belgian numbers</a>.
%H A106039 Eric Angelini, <a href="/A106039/a106039.pdf">Belgian Numbers</a> [Cached copy with permission]
%e A106039 13 is a Belgian-0 number because of the sequence
%e A106039 0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, ...
%e A106039 the first differences of which are
%e A106039 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
%e A106039 176 is a Belgian-0 number because, starting from 0 (the seed), one can build a sequence containing 176 in this way:
%e A106039 0.1.8.14.15.22.28.29.36.42.43.50.....155.162.168.169.176.... (sequence)
%e A106039 .1.7.6..1..7..6..1..7..6..1..7..........7...6...1...7.. (first differences)
%e A106039 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.
%t A106039 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 *)
%o A106039 (Haskell)
%o A106039 a106039 n = a106039_list !! (n-1)
%o A106039 a106039_list = filter belge0 [0..] where
%o A106039    belge0 n = n == (head $ dropWhile (< n) $
%o A106039                     scanl (+) 0 $ cycle ((map (read . return) . show) n))
%o A106039 -- _Reinhard Zumkeller_, May 07 2015
%Y A106039 Cf. A257782 (complement), A253717 (primes).
%Y A106039 Belgian-k numbers, k=0..9: A106039, A106439, A106518, A106596, A106631, A106792, A107014, A107018, A107032, A107043.
%Y A106039 Cf. A257770, A257778.
%K A106039 base,easy,nonn
%O A106039 1,3
%A A106039 _Eric Angelini_, Jun 07 2005
%E A106039 Offset changed by _Reinhard Zumkeller_, May 08 2015