Hello Math-Fun [and Jean Fontaine], I've coined "Schmitter sequences" yesterday because I had to write a little something for a friend's birthday (don't spread elsewhere msg and links until sept. 29th please) A Schmitter seq is a seq of positive integers, none of which is re- peated, built on a name, a word, a sentence, etc. Example (in English) with "Fred Schmitter" (paper in French there http://www.cetteadressecomportecinquantesignes.com/Schmitter.htm ): Two simple rules : - reproduce ad infinitum the block FREDSCHMITTER - replace one by one each letter with the smallest integer not already used in the sequence and containing (in english) the said letter. So, the first FREDSCHMITTER block produces: Four 4 zeRo 0 onE 1 one hunDred 100 Six 6 one oCtillion 10^27 tHree 3 one Million 10^6 fIve 5 Two 2 eighT 8 ninE 9 thiRteen 13 See: "Spelling n in English requires a new letter of the alphabet." http://www.research.att.com/~njas/sequences/A050933 0, 1, 2, 3, 4, 5, 6, 8, 11, 20, 100, 101, 1000000, 1000000000, 1000000000000000, 1000000000000000000000000, 1000000000000000000000000000 The large numbers are (new letter in uppercase): "one Billion" (10^9), "one Quadrillon" (10^15), "one sePtillion" (10^24) and "one oCtillion" (10^27). Their names and definitions are based on the American system. and "Number of distinct letters needed to spell English names of numbers 1 through n" http://www.research.att.com/~njas/sequences/A082655 3, 5, 7, 9, 11, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16 -- Late increases are at twentY, hunDred, thousAnd, Million, Billion, Quadrillion, sePtillion, oCtillion. Only J and K are never used for English number names. Z is used only for zero. The second FREDSCHMITTER block will be: Fourteen 14 twenty-thRee 23 sEven 7 one hunDred one 101 Sixteen 16 one oCtillion one 10^27 + 1 eigHteen 18 one Million one 10^6 + 1 fIfteen 15 Ten 10 Twelve 12 Eleven 11 twenty-fouR 24 etc. The seq arising from "Fred Schmitter" is the concatenation of all blocks -- and starts like this (if I didn't make too many mistakes -- but you have the general idea now): S = 4,0,1,100,6,10^27,3,10^6,5,2,8,9,13,14,23,7,101,16,... Question: What is the smallest (english spelled) number yielding to a "complete Schmitter sequence" -- a sequence which is the re- arrangement of the complete set of integers? In French it is 15 (QUINZE). And what are the next such "Schmitter numbers"? I've more questions (see the French link above), but my wife desperately asks me to step out for a movie... Best, É.