A107070 Numbers m with the following property. Suppose m = 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 m.

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

Offset: 1

Author

Eric Angelini, Jun 07 2005

Keywords

• base
• nonn

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

Examples

			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)
		

Links

• E. Angelini, Belgian numbers.
• E. Angelini, Belgian Numbers [Cached copy with permission]
• J.-P. Davalan, Nombres belges [Includes applets to generate sequence]

Crossrefs

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

Programs

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 *)

Extensions

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