Jim Nastos nastos at gmail.com
Sat Dec 26 23:28:14 CET 2009

```Hi,

For your observation, the relevant properties in effect are

(1) if a(i) is 0 mod 9, then a(i) plus or minus its digit-sum is also
0 mod 9. (So once a trajectory goes into that congruence class, it
never leaves.)

(2) performing a subtraction operation will always put the next term
into the 0 mod 9 congruence class

(3) a cycle will has at least one subtraction step.

JN

On Sat, Dec 26, 2009 at 1:49 PM, A.N.W.Hone <A.N.W.Hone at kent.ac.uk> wrote:
> Hi seqfans,
>
> I haven't tried to prove this, but a simple remark: all the terms in the examples of cycles are congruent to 0 mod 9.
>
> Andy
> ________________________________________
> From: seqfan-bounces at list.seqfan.eu [seqfan-bounces at list.seqfan.eu] On Behalf Of zak seidov [zakseidov at yahoo.com]
> Sent: 26 December 2009 16:08
> To: seqfaneu
>
>
> Dear seqfans,
>
> Does the sequence defines by recurrence
> a(n+1)=a(n)+/-sd(a(n)), if  a(n) is odd/even,
> with sd(m)=sum of digits of m,
> end in cycle for any initial a(1)?
>
> Here are 4 examples with cycles of various lengths.
>
> At a(1)=1, the sequence is:
> 1,2,0,0,0,0,0,0
> with cycle 0, and the next term is 0.
>
> At a(1)=5, the sequence is:
> 5,10,9,18,9,18,9,18
> with cycle 9,18, and the next term is 9.
>
> At a(1)=1711, the sequence is:
> 1711,1721,1732,1719,1737,1755,1773,1791,1809,1827,1845,1863,1881,1899,1926,1908,1890,1872,1854,1836,1818,1800,1791,1809,1827,1845,1863,1881,1899,1926
> with cycle 1791,1809,1827,1845,1863,1881,1899,1926,
>  1908,1890,1872,1854,1836,1818,1800,
> and the next term is 1791.
>
> At a(1)=10810065, the sequence is:
> 10810065,10810086,10810062,10810044,10810026,10810008,10809990,10809954,10809918,10809882,10809846,10809810,10809783,10809819,10809855,10809891,
> 10809927,10809963,10809999,10810044,10810026,10810008,10809990,10809954,
>  with cycle   10810044,10810026,10810008,10809990,10809954,
> 10809918,10809882,10809846,10809810,10809783,10809819,10809855,10809891,10809927,10809963,10809999,
> and the next term is 10810044.
>
> Merry Christmas and happy New Year to ALCORN!
> Zak
>
>
>
>
>
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>
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>
>
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>

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