[seqfan] Re: Self-stuffable numbers

[M. F. Hasler]

On 2 Jan 2019 12:15, "Hans Havermann" <gladhobo at bell.net> wrote:

MFH: "The main challenge is to find larger (primitive) roots, i.e., terms
of A322323 (without repetitions and) with trailing zeros removed.
Arbitrarily many larger terms of A322323 are then computed
straightforwardly."

Lars Blomberg has found 210210210210210210210 to be self-stuffable. While
it is trivially true that it could have been computed from the primitive
root 21021021021021021021, I don't believe that a b-file for A322002 with
terms that large is in the offing.


Such "sporadic" solutions are unavoidable.
For  n = 210  with  S(n) = 221100 we have S(n) = -30 (mod 210)
and more generally for
n_m = n * R_m(10^3),  R_m = (X^m - 1)/(X - 1)
we have S(n_m) = R_m(10^6) * S(n) = - 30 m  (mod 210)
i.e.,
S( R_2(10^3) * 210 ) = S( 210 210 ) = 150  (mod 210)
S( R_3(10^3) * 210 ) = S( 210 210 210  ) = 120  (mod 210)
...
and finally  S( R_7(10^3)*210 ) = 0 (mod 210)
and it turns out that also
 R_7(10^3)*210  |  R_7(10^6)*221100
so that  210_7 = 210 210 210 210 210 210 210 is also self-stuffable
(and of course this is also the case for 7 replaced by any odd multiple of
7).
I try to formulate a general sufficient condition for some (probably always
odd) number of concatenations of n to be eventually self-stuffable (and
thus yield a primitive root).

- Maximilian
PS: I use the occasion to acknowledge improvements due to Ray, in my
comment in A322323 (clarification M | S(M)) and example in A322002.