OFFSET
1,6
COMMENTS
Conjecture: for n>=3, a(n)>0.
Records are 1,14,23,50,252,4752,...
The corresponding primes are 2,13,131,653,883,1279,...
These primes beginning with the second one we call "stubborn primes".
Counter-conjecture: a(2889)=0. - Hans Havermann, Oct 15 2014
If a(n)=1, then the resulting primes are in A092993 and form A055782; if a(n)=2, then they form sequence 4133,4733,5333,7933,..., etc. - Vladimir Shevelev, Oct 16 2014
If a prime p divides Pb_k, then it also divides Pb_{k+m(p-1)} for all m>=0. This follows from Fermat's little theorem applied to b_x=(10^x-1)/3 with x=p-1. - M. F. Hasler, Oct 20 2014
LINKS
Hans Havermann, Table of n, a(n) for n = 1..2888 (first 200 terms from Michel Marcus)
Hans Havermann, "Google+ discussion relating to this sequence"
Vladimir Shevelev, "Stubborn primes"
EXAMPLE
For n=1, start with prime(1)=2 and get already at the first step the prime 23. So a(1)=1.
For n=2, starting with prime(2)=3, one never gets a prime by appending further digits "3", therefore a(2)=0.
For n=3, n=4, n=5, one gets after the first step the primes 53, 73, 113, and therefore a(n)=1.
For n=6, start with prime(6)=13; one has to append 14 "3"s in order to get a new prime, so a(6)=14.
For n=2889, start with prime(2889) = 26293. (Do not mix up with prime(2899) = 26393...!) Appending 2k-1 or 6k-4 or 6k-2 or 18k-6 or 36k-18 or 180k-144 digits "3" yields a number divisible by 11 resp. 7 resp. 13 resp. 19 resp. 101 resp. 31. For 18k-12 and 36k (with k <> 1 (mod 5)) digits "3" there is no simple pattern and both yield sometimes large primes in the factorization, but (so far) always composite numbers 26293...3 (up to several thousand digits). - M. F. Hasler, Oct 16 2014
MATHEMATICA
f[n_] := Block[{k = 1, p = Prime@ n}, While[ !PrimeQ[p*10^k + (10^k - 1)/3], k++]; k]; f[2] = 0; Array[f, 100] (* Robert G. Wilson v, Apr 24 2015 *)
m3[n_]:=Module[{k=10n+3}, While[!PrimeQ[k], k=10k+3]; IntegerLength[k]-IntegerLength[ n]]; Join[{1, 0}, m3/@Prime[Range[3, 90]]] (* Harvey P. Dale, Feb 11 2018 *)
PROG
(PARI) a(n) = {if (n==2, return (0)); p = prime(n); k = 1; while (! isprime(p = p*10+3), k++); k; } \\ Michel Marcus, Sep 13 2014
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Vladimir Shevelev, Sep 13 2014
EXTENSIONS
More terms from Peter J. C. Moses, Sep 13 2014
STATUS
approved