

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
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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 quasiautomorphic sequence. Thus this sequence is infinite.  Robert G. Wilson v, May 06 2011
The existence of the nine templates upon which the quasiautomorphic sequences are decided guarantees that no more than nine solutions exist for a given digitlength. The equidistribution of the ten baseten digits within these templates predicts a longterm average of two solutions per digitlength. All nine solutions happen trivially for digitlength 1 (terms 19) and not again until digitlength 1899283 (terms 35947283594736).  Hans Havermann, May 27 2011, Aug 15 2011
The nth 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: A302500 A102493 A024661 * A320081 A243507 A243023
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



