%I #35 Dec 05 2019 10:45:26

%S 1,3,3,2,0,1,3,2,2,1,3,4,4,3,3,3,3,2,3,3,4,2,2,5,4,4,2,5,5,4,4,2,2,5,

%T 4,4,2,5,5,4,2,3,3,4,2,2,3,4,4,2,2,3,3,4,2,2,3,4,4,2,2,3,3,4,2,2,3,4,

%U 4,2,3,4,4,5,5,5,4,5,5,5,4,2,2,5,4,4,2,5,5,4,4,2,2,5,4,4,2,5,5,4

%N Map numbers to number of letters in English name; sequence gives number of steps to converge (to 4).

%C The smallest n with a(n) = 7 is 1103323373373373373373373373373 (one nonillion one hundred and three octillion ...) which has 323 letters. - _Roland Kneer_, Jul 04 2013

%C From _Robert G. Wilson v_, Mar 13 2017: (Start)

%C First occurrence of k = 0,1,2,...: 4, 0, 3, 1, 11, 23, 323, 1103323373373373373373373373373, etc.;

%C 0 only occurs at 4;

%C 1 only occurs for n = 0, 5 & 9;

%C 2 only occurs for n = 3, 7, 8, 17, 21, 22, 26, 31, 32, 36, 40, 44, 45, 49, 50, 54, 55, 59, 60, 64, 65, 69, 81, 82, 86, 91, 92 & 96;

%C 3 occurs for n = 1, 2, 6, 10, 13, 14, 15, 16, 18, 19, 41, 42, 46, 51, 52, ..., ;

%C 4 occurs for n = 11, 12, 20, 24, 25, 29, 30, 34, 35, 39, 43, 47, 48, 53, ..., ;

%C 5 occurs for n = 23, 27, 28, 33, 37, 38, 73, 74, 75, 77, 78, 79, 83, 87, ..., ;

%C 6 occurs for n = 323, 327, 328, 333, 337, 338, 374, 375, 379, 383, 387, ..., ; etc.

%C (End)

%C The basis of this sequence is that integers are named without the use of "and". As a minor correction at the time of this comment, therefore, the 31-digit number above beginning 1103 ... should be described as "one nonillion one hundred three octillion ...". If naming is done in accordance with UK English usage ("one hundred", "one hundred and one", ...), the first occurrence of a(n) = 0, 1, 2, 3, 4, 5, 6, 7, ... is for n = 4, 0, 3, 1, 11, 23, 124, 113373373373, ... _Ian Duff_, Dec 04 2019

%e 1 -> 3 -> 5 -> 4, so a(1) = 3.

%t (* get t from A005589 *) f[n_] := Length@ NestWhileList[ StringLength@ t[[# + 1]] &, n, UnsameQ, 2] - 2; Array[f, 100, 0] (* _Robert G. Wilson v_, Jun 01 2012 *)

%K nonn,word

%O 0,2

%A _Robert G. Wilson v_

%E Corrected at the suggestion of _Kevin Ryde_ by _Robert G. Wilson v_, Jun 01 2012