A109776 - OEIS

%I #14 Mar 30 2024 21:20:23

%S 22,4444,224444,442244,444422,666666,10123133,10123331,10143133,

%T 10143331,10153133,10153331,10163133,10163331,10173133,10173331,

%U 10183133,10183331,10193133,10193331,10212332,10213223,10232132

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

%C From _Robert G. Wilson v_, May 05 2012: (Start)

%C 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.

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

%C 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.

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

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

%C (End)

%H Robert G. Wilson v, <a href="/A109776/b109776.txt">Table of n, a(n) for n = 1..10538</a>

%H The Prime Puzzles & Problems Connection by Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_324.htm">Puzzle"> 324. Self-descriptive numbers</a>.

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

%t 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 *)

%Y Cf. A108810, A173101, A005150.

%K nonn,base,fini

%O 1,1

%A _Jud McCranie_, Aug 15 2005