# [seqfan] Re: "Stubborn primes"

M. F. Hasler seqfan at hasler.fr
Sun Sep 14 16:29:26 CEST 2014

```On Sat, Sep 13, 2014 at 5:05 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> Only number 1333333333333333 was prime. So I call 13 a "stuborn prime". After
> that with help Peter I have submitted to-day  sequence A232210. The calculated records of numbers of concatunating 3's are 1,14,23,50,252,4752 with the
> correspoding primes 5,13,131,653, 883,1279,...

IMHO, some more "fundamental" (if I dare...) variants of this and
related sequences (xrefs in A092993 ; xrefs are missing in A232210!)
[the latter being just the obvious subsequence obtianed by composition
with A40] :

a1(n) = least number k>0 of "1" to be appended to n in order to get a prime
a3(n) = least number k>0 of "3" to be appended to n in order to get a prime*
a7(n) = least number k>0 of "7" to be appended to n in order to get a prime
a9(n) = least number k>0 of "9" to be appended to n in order to get a prime*

* or 0 if no such k exists

b[X](n) = least number k>=0 of "X" to be appended to n in order to get a prime

** or -1 if no such k exists

c[X] ; d[X] : the resulting primes (or 0 if no such prime exists)

e[X] (and others ?) : records

[(ii) - variants of A092993]
f[X](n) = least number k>0 such that k concatenations of n followed by
"X" yields a prime*
g[X] : the resulting primes

PS: for the PARI code the construct p=p*10+X is more efficient than
p=eval(Str(p,X))

Regards & a nice Sunday to all SeqFans,
Maximilian

```