[seqfan] Fibonaroot

Eric Angelini Eric.Angelini at kntv.be
Sat Jul 13 17:01:50 CEST 2013

Hello SeqFans,

Consider the set P* as the union of A000040 (the prime numbers)
and A028835 (numbers with a prime Digital Root -- the DR of the
integer N being the iterated sum of the digits of N).

The sequence S(1) herunder has a(1)=1 and a(2)=1 (the seed);
then, from a(3) on, we have for a(n) the sum of the two previous
terms of S.

With a twist.

The twist being that when the said sum does NOT belong to P*, we
replace it with its digital root (DR). Let's see how S starts:


All terms > 9 are part of P* but now the sum 21+34 is not (=55);
we replace thus 55 by its DR, which is 1:


The sum 34+1 is not part of P* (=35); we replace it by its DR (=8):

We proceed mechanically like this -- and quickly enter into a loop:


If we call S(k) a sequence with the seed [1,k], we see that S(2) ends
also in a loop:

S(2)=1,2,3,... (as the fate of a sequence containing two consecutive
                terms already seen in a previous sequence is known).


I guess S(3) and S(4) will end soon in a loop too -- but is there a
S(k) sequence which doesn't?

[Please forgive me if this is old hat -- and/or filled with typos, as

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