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A107070 Numbers n with the following property. Suppose n = 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 n. 1

%I #40 Jun 09 2016 11:08:40

%S 1,2,3,4,5,6,7,8,9,61,71,918,3612,5101,8161,12481,51011,248161,361213,

%T 5101111,7141519,8161723,481617232,2481617232,4816172324,5101111121,

%U 24816172324,51011111213,71415192025,612131516192,816172324313,3612131516192,5101111121314,6121315161920,9181927283739

%N Numbers n with the following property. Suppose n = 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 n.

%C These are sometimes called Eric numbers or Belgian numbers. - _N. J. A. Sloane_, May 06 2011

%C Each digit {1..9} will produce a quasi-automorphic sequence. Thus this sequence is infinite. - _Robert G. Wilson v_, May 06 2011

%C 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

%C 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

%H E. Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/EricNumbers.htm">Belgian numbers</a>.

%H E. Angelini, <a href="/A106039/a106039.pdf">Belgian Numbers</a> [Cached copy with permission]

%H J.-P. Davalan, <a href="http://jeux-et-mathematiques.davalan.org/mots/suites/belge/index.html">Nombres belges</a> [Includes applets to generate sequence]

%e The following example shows why 61 is a member:

%e 6.12.13.19.20.26.27.33.34.40.41.47.48.54.55.61... (sequence)

%e .6..1..6..1..6..1..6..1..6..1..6..1..6..1..6... (first differences)

%t 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 *)

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

%K base,nonn

%O 1,2

%A _Eric Angelini_, Jun 07 2005

%E Minor edits by _N. J. A. Sloane_, May 06 2011

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Last modified March 28 04:05 EDT 2024. Contains 371235 sequences. (Running on oeis4.)