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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
1, 2, 3, 4, 5, 6, 7, 8, 9, 61, 71, 918, 3612, 5101, 8161, 12481, 51011, 248161, 361213, 5101111, 7141519, 8161723, 481617232, 2481617232, 4816172324, 5101111121, 24816172324, 51011111213, 71415192025, 612131516192, 816172324313, 3612131516192, 5101111121314, 6121315161920, 9181927283739 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

These are sometimes called Eric numbers or Belgian numbers. - N. J. A. Sloane, May 06 2011

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

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

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

LINKS

Table of n, a(n) for n=1..35.

E. Angelini, Belgian numbers.

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

J.-P. Davalan, Nombres belges [Includes applets to generate sequence]

EXAMPLE

The following example shows why 61 is a member:

6.12.13.19.20.26.27.33.34.40.41.47.48.54.55.61... (sequence)

.6..1..6..1..6..1..6..1..6..1..6..1..6..1..6... (first differences)

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A070244 A102493 A024661 * A243507 A243023 A243024

Adjacent sequences:  A107067 A107068 A107069 * A107071 A107072 A107073

KEYWORD

base,nonn

AUTHOR

Eric Angelini, Jun 07 2005

EXTENSIONS

Minor edits by N. J. A. Sloane, May 06 2011

STATUS

approved

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Last modified May 25 08:37 EDT 2017. Contains 287015 sequences.