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 A106039 Belgian-0 numbers. 20
 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 (list; graph; refs; listen; history; text; internal format)
 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: 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

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Last modified December 11 19:37 EST 2018. Contains 318049 sequences. (Running on oeis4.)