A109776 Self-describing numbers: reading the number gives a (possibly redundant) description of the number.

22, 4444, 224444, 442244, 444422, 666666, 10123133, 10123331, 10143133, 10143331, 10153133, 10153331, 10163133, 10163331, 10173133, 10173331, 10183133, 10183331, 10193133, 10193331, 10212332, 10213223, 10232132

Offset: 1

Author

Jud McCranie, Aug 15 2005

Keywords

• nonn
• base
• fini

Comments

From Robert G. Wilson v, May 05 2012: (Start)

If abcd... with a, b, c & d integers, then so is cdab... . As an example, since 10123133 is a term so must be 10123331, 10311233, 10313312, 10331231, 10333112, 12103133, 12103331, 12311033, 12313310, 12331031, 12333110, 31101233, 31103312, 31121033, 31123310, 31331012, 31331210, 33101231, 33103112, 33121031, 33123110, 33311012, 33311210.

Therefore 10123133 can be said to be the progenerator or the primitive self-describing number.

Also if we index the number abcd... from left to right, the sum of the odd indexes must equal the number of digits for unique even-indexed digits.

Number of terms < 10^2n: 1, 2, 6, 1043, 5498, ..., .

This sequence is finite with the last term is probably 9998979595959595848484848484848476737373737373736262626262625151515110.

(End)

Examples

			"22" does indeed consist of "two 2's".
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10538
• The Prime Puzzles & Problems Connection by Carlos Rivera, Puzzle"> 324. Self-descriptive numbers.

Crossrefs

Cf. A108810, A173101, A005150.

Programs

Mathematica

fQ[n_] := Block[{id = IntegerDigits[n]}, If[ OddQ[ Length[id]], Return[False], Union[Reverse@# & /@ Tally[id]] == Union@ Partition[id, 2]]]; k = 1; lst = {}; While[k < 10^7, If[fQ@ k, AppendTo[lst, k]; Print[k]]; k++]; lst (* Robert G. Wilson v, Apr 27 2012 *)