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
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
KEYWORD
base,nonn
AUTHOR
Eric Angelini, Jun 07 2005
EXTENSIONS
Minor edits by N. J. A. Sloane, May 06 2011
STATUS
approved