# [seqfan] Re: Add-if-odd sequence

A.N.W.Hone A.N.W.Hone at kent.ac.uk
Sat Dec 26 22:49:16 CET 2009

```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
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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
Subject: [seqfan]  Add-if-odd sequence

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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