Placid Platypus

The Placid Platypus function $PP(n)$ is the minimal number of states of 2-symbol Turing machines that print exactly n ones when halt.

$PP(n)\le n$, because you can print each 1 in a state. Also if $n>\mathrm{\Sigma }(m)$, then $PP(n)>m$. On the other side, there is also a connection to Lazy Beaver (or, a variant that is instead determined by ones rather than steps): if $n<LB(m)$ then $PP(n)\le m$.

Note that this function is not strictly increasing, and that it is computable.

Values

n	$PP(n)$	Example TM
1	1	1RZ--- (bbch)
2	2	1RB---_1RZ--- (bbch)
3	2	1RB1LA_1LA1RZ (bbch)
4	2	1RB1LB_1LA1RZ (bbch)
5	3	1RB1RZ_1LB0RC_1LC1LA (bbch)
6	3	1RB1RZ_0RC1RB_1LC1LA (bbch)
7	4	1RB1LC_0RC0RB_0LD0LA_1LA1RZ (bbch)
8	4	1RB1RZ_1LC0RD_1LA1LB_0LC1RD (bbch)
9	4	1RB1LD_1LC0RB_1RA1LA_1RZ0LC (bbch)
10	4	1RB1RZ_1LC1RA_0RC0LD_1RD0LB (bbch)
11	4	1RB1LD_0LC0RC_1LC1LA_1RZ0LA (bbch)
12	4	1RB0RD_1LC0LA_1RA1LB_1RZ0RC (bbch)
13	4	1RB1LB_1LA0LC_1RZ1LD_1RD0RA (bbch)
14	5	1RB0LB_0LC1RZ_1LE0RD_1RC0RA_1LD0LB (bbch)
15	5	1RB0LD_1LC0LB_1RD0RC_1LA0RE_1RA1RZ (bbch)