Earthworm Sequence/Examples

Examples of Earthworm Sequence
The earthworm sequence is analysed for all $n \in \set {00, 01, 02, \dotsc, 99}$, not only for the range $\set {10, 11, 12, \dotsc, 99}$.

The first table lists the integers that end up in a loop of $20$ terms:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $\map h n$ ! align="right" style = "padding: 2px 10px" | $\map h {\map h n}$ ! align="right" style = "padding: 2px 10px" | $\map h {\map h {\map h n} }$
 * align="right" style = "padding: 2px 10px" | $01$, $51$
 * align="right" style = "padding: 2px 10px" | $02$
 * align="right" style = "padding: 2px 10px" | $04$
 * align="right" style = "padding: 2px 10px" | $08$
 * align="right" style = "padding: 2px 10px" | $27$, $77$
 * align="right" style = "padding: 2px 10px" | $54$
 * align="right" style = "padding: 2px 10px" | $08$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $29$, $79$
 * align="right" style = "padding: 2px 10px" | $58$
 * align="right" style = "padding: 2px 10px" | $16$
 * align="right" style = "padding: 2px 10px" | $32$
 * align="right" style = "padding: 2px 10px" | $33$, $83$
 * align="right" style = "padding: 2px 10px" | $66$
 * align="right" style = "padding: 2px 10px" | $32$
 * align="right" style = "padding: 2px 10px" | $64$
 * align="right" style = "padding: 2px 10px" | $41$, $91$
 * align="right" style = "padding: 2px 10px" | $82$
 * align="right" style = "padding: 2px 10px" | $64$
 * align="right" style = "padding: 2px 10px" | $28$
 * align="right" style = "padding: 2px 10px" | $07$, $57$
 * align="right" style = "padding: 2px 10px" | $14$
 * align="right" style = "padding: 2px 10px" | $28$
 * align="right" style = "padding: 2px 10px" | $56$
 * align="right" style = "padding: 2px 10px" | $39$, $89$
 * align="right" style = "padding: 2px 10px" | $78$
 * align="right" style = "padding: 2px 10px" | $56$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $03$, $53$
 * align="right" style = "padding: 2px 10px" | $06$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $24$
 * align="right" style = "padding: 2px 10px" | $31$, $81$
 * align="right" style = "padding: 2px 10px" | $62$
 * align="right" style = "padding: 2px 10px" | $24$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $37$, $87$
 * align="right" style = "padding: 2px 10px" | $74$
 * align="right" style = "padding: 2px 10px" | $48$
 * align="right" style = "padding: 2px 10px" | $96$
 * align="right" style = "padding: 2px 10px" | $49$, $99$
 * align="right" style = "padding: 2px 10px" | $98$
 * align="right" style = "padding: 2px 10px" | $96$
 * align="right" style = "padding: 2px 10px" | $92$
 * align="right" style = "padding: 2px 10px" | $23$, $73$
 * align="right" style = "padding: 2px 10px" | $46$
 * align="right" style = "padding: 2px 10px" | $92$
 * align="right" style = "padding: 2px 10px" | $84$
 * align="right" style = "padding: 2px 10px" | $21$, $71$
 * align="right" style = "padding: 2px 10px" | $42$
 * align="right" style = "padding: 2px 10px" | $84$
 * align="right" style = "padding: 2px 10px" | $68$
 * align="right" style = "padding: 2px 10px" | $17$, $67$
 * align="right" style = "padding: 2px 10px" | $34$
 * align="right" style = "padding: 2px 10px" | $68$
 * align="right" style = "padding: 2px 10px" | $36$
 * align="right" style = "padding: 2px 10px" | $09$, $59$
 * align="right" style = "padding: 2px 10px" | $18$
 * align="right" style = "padding: 2px 10px" | $36$
 * align="right" style = "padding: 2px 10px" | $72$
 * align="right" style = "padding: 2px 10px" | $43$, $93$
 * align="right" style = "padding: 2px 10px" | $86$
 * align="right" style = "padding: 2px 10px" | $72$
 * align="right" style = "padding: 2px 10px" | $44$
 * align="right" style = "padding: 2px 10px" | $11$, $61$
 * align="right" style = "padding: 2px 10px" | $22$
 * align="right" style = "padding: 2px 10px" | $44$
 * align="right" style = "padding: 2px 10px" | $88$
 * align="right" style = "padding: 2px 10px" | $47$, $97$
 * align="right" style = "padding: 2px 10px" | $94$
 * align="right" style = "padding: 2px 10px" | $88$
 * align="right" style = "padding: 2px 10px" | $76$
 * align="right" style = "padding: 2px 10px" | $19$, $69$
 * align="right" style = "padding: 2px 10px" | $38$
 * align="right" style = "padding: 2px 10px" | $76$
 * align="right" style = "padding: 2px 10px" | $52$
 * align="right" style = "padding: 2px 10px" | $13$, $63$
 * align="right" style = "padding: 2px 10px" | $26$
 * align="right" style = "padding: 2px 10px" | $52$
 * align="right" style = "padding: 2px 10px" | $04$
 * }
 * align="right" style = "padding: 2px 10px" | $11$, $61$
 * align="right" style = "padding: 2px 10px" | $22$
 * align="right" style = "padding: 2px 10px" | $44$
 * align="right" style = "padding: 2px 10px" | $88$
 * align="right" style = "padding: 2px 10px" | $47$, $97$
 * align="right" style = "padding: 2px 10px" | $94$
 * align="right" style = "padding: 2px 10px" | $88$
 * align="right" style = "padding: 2px 10px" | $76$
 * align="right" style = "padding: 2px 10px" | $19$, $69$
 * align="right" style = "padding: 2px 10px" | $38$
 * align="right" style = "padding: 2px 10px" | $76$
 * align="right" style = "padding: 2px 10px" | $52$
 * align="right" style = "padding: 2px 10px" | $13$, $63$
 * align="right" style = "padding: 2px 10px" | $26$
 * align="right" style = "padding: 2px 10px" | $52$
 * align="right" style = "padding: 2px 10px" | $04$
 * }
 * align="right" style = "padding: 2px 10px" | $52$
 * align="right" style = "padding: 2px 10px" | $04$
 * }

The second table lists the integers that end up in a loop of $4$ terms:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $\map h n$ ! align="right" style = "padding: 2px 10px" | $\map h {\map h n}$ ! align="right" style = "padding: 2px 10px" | $\map h {\map h {\map h n} }$
 * align="right" style = "padding: 2px 10px" | $05$, $55$
 * align="right" style = "padding: 2px 10px" | $10$
 * align="right" style = "padding: 2px 10px" | $20$
 * align="right" style = "padding: 2px 10px" | $40$
 * align="right" style = "padding: 2px 10px" | $35$, $85$
 * align="right" style = "padding: 2px 10px" | $70$
 * align="right" style = "padding: 2px 10px" | $40$
 * align="right" style = "padding: 2px 10px" | $80$
 * align="right" style = "padding: 2px 10px" | $45$, $95$
 * align="right" style = "padding: 2px 10px" | $90$
 * align="right" style = "padding: 2px 10px" | $80$
 * align="right" style = "padding: 2px 10px" | $60$
 * align="right" style = "padding: 2px 10px" | $15$, $65$
 * align="right" style = "padding: 2px 10px" | $30$
 * align="right" style = "padding: 2px 10px" | $60$
 * align="right" style = "padding: 2px 10px" | $20$
 * }
 * align="right" style = "padding: 2px 10px" | $30$
 * align="right" style = "padding: 2px 10px" | $60$
 * align="right" style = "padding: 2px 10px" | $20$
 * }

The second table lists the integers that end up in a loop of $1$ term, that is, the degenerate case that ends in $0$:


 * {| border="1"


 * align="right" style = "padding: 2px 10px" | $25$, $75$
 * align="right" style = "padding: 2px 10px" | $50$
 * align="right" style = "padding: 2px 10px" | $00$
 * align="right" style = "padding: 2px 10px" | $00$
 * }
 * }