[seqfan] Re: A202286
Hans Havermann
gladhobo at teksavvy.com
Wed Mar 14 18:42:54 CET 2012
> Isn't this sequence finite?
M. F. Hasler, Mar 13 2012, in his A202286 edit: "The sequence is
finite, more specifically there cannot be more than 28 terms. Proof:
To have p = k*R(p) -/+ q, with q < p, we must have 0 < k <= (p +/-
q) / R(p) < 2p / (p/10) = 20, since the prime p cannot end in 0 and
therefore R(p) > p/10 (because p and R(p) have the same number of
digits). So we must have 1 <= k <= 19 in the p+q case, and 1 <= k <= 9
in the p-q case."
Isn't maximum k the sole determinant of the number of terms in this
sequence? If there was a 20-ways term wouldn't there have to be a k>19
in order for this to occur?
I think the separation into +q and -q cases isn't very helpful here:
If p<R(p), k can only be 1 so, generally, p>R(p) which immediately
suggests at most 9 +q's. If that were ever to happen, it would have to
happen for k=1..9 with an additional of at most 10 -q's (for
k=10..19). Of course k can be <10 for some -q but that reduces the
number of available +q's.
Here are my eight {k,q} for p=845534401:
{3,532227757}
{5,323356661}
{6,218921113}
{8,10050017}
{9,-94385531}
{11,-303256627}
{14,-616563271}
{15,-720998819}
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