%I #26 Jun 26 2022 10:48:03

%S 0,1,1,2,1,2,1,2,2,2,1,2,1,2,1,3,1,2,1,2,2,2,1,2,2,2,2,3,1,3,1,2,1,2,

%T 1,3,1,2,2,3,1,3,1,3,2,2,1,3,2,2,1,2,1,2,2,2,2,2,1,2,1,2,3,2,1,3,1,2,

%U 1,3,1,3,1,2,2,3,1,3,1,3,2,2,1,2,1,2,1,3,1,3,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2

%N Number of iterations of the map k -> A000001(k) needed to reach 1 starting at n, or -1 if no such number exists.

%C The old entry with this sequence number was a duplicate of A052409.

%C It appears that a(n) is the number of times n appears in A142978, excluding the first column of infinitely many 1's. - _Ron Wolf_, Dec 16 2020

%C Preceding comment is incorrect. The first counterexample is a(19) = 1, whereas 19 appears twice in A142978. - _Eric M. Schmidt_, Mar 22 2021

%H Eric M. Schmidt, <a href="/A072410/b072410.txt">Table of n, a(n) for n = 1..2047</a> [using data from example and A000001. a(1024) corrected by _Jinyuan Wang_, Jun 26 2022]

%H John H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>.

%e Conway et al. remark that every number less than 2048 reaches 1 after at most 5 steps and give the following examples:

%e 672 -> 1280 -> 1116461 -> 1

%e 1024 -> 49487367289 -> 1

%e 720 -> 840 -> 186 -> 6 -> 2 -> 1

%e 320 -> 1640 -> 68 -> 5 -> 1

%e 384 -> 20169 -> 67 -> 1

%e 128 -> 2328 -> 64 -> 267 -> 1

%e 960 -> 11394 -> 60 -> 13 -> 1

%e 864 -> 4725 -> 51 -> 1

%e 1344 -> 11720 -> 49 -> 2 -> 1

%e 1440 -> 5958 -> 16 -> 14 -> 2 -> 1

%e 1248 -> 1460 -> 15 -> 1

%e 256 -> 56092 -> 11 -> 1

%e 1728 -> 47937 -> 6 -> 2 -> 1

%e 512 -> 10494213 -> 5 -> 1

%e 1536 -> 408641062 -> 4 -> 2 -> 1

%e 1664 -> 21507 -> 2 -> 1

%e 1280 -> 1116461 -> 1

%Y Cf. A000001, A066952 (indices of records).

%K nonn

%O 1,4

%A _N. J. A. Sloane_, Oct 03 2008