This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A232210 #87 Dec 23 2024 14:53:43
%S A232210 1,0,1,1,1,14,1,1,1,1,1,1,2,1,2,2,1,1,1,3,1,2,6,2,2,1,1,2,1,4,4,23,1,
%T A232210 2,1,6,2,2,5,1,10,2,1,1,1,1,1,6,1,1,1,1,2,4,2,1,1,1,2,4,1,2,5,4,2,3,1,
%U A232210 1,5,4,1,1,1,1,1,1,2,2,1,1,6,4,2,14,2,4,1,3
%N A232210 Let b_k=3...3 consist of k>=1 3's. Then a(n) is the smallest k such that the concatenation prime(n)b_k is prime, or a(n)=0 if there is no such prime.
%C A232210 Conjecture: for n>=3, a(n)>0.
%C A232210 Records are 1,14,23,50,252,4752,...
%C A232210 The corresponding primes are 2,13,131,653,883,1279,...
%C A232210 These primes beginning with the second one we call "stubborn primes".
%C A232210 Counter-conjecture: a(2889)=0. - _Hans Havermann_, Oct 15 2014
%C A232210 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
%C A232210 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
%H A232210 Hans Havermann, <a href="/A232210/b232210.txt">Table of n, a(n) for n = 1..2888</a> (first 200 terms from Michel Marcus)
%H A232210 Hans Havermann, <a href="https://plus.google.com/u/0/101745241027004457169/posts/TUvdTWuz6y2">"Google+ discussion relating to this sequence"</a>
%H A232210 Vladimir Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-September/013620.html">"Stubborn primes"</a>
%H A232210 Robert G. Wilson v, <a href="/A232210/a232210.txt">Table of n, a(n) for n = 1..10000 with -1 for those entries where a(n) has not yet been found</a>
%e A232210 For n=1, start with prime(1)=2 and get already at the first step the prime 23. So a(1)=1.
%e A232210 For n=2, starting with prime(2)=3, one never gets a prime by appending further digits "3", therefore a(2)=0.
%e A232210 For n=3, n=4, n=5, one gets after the first step the primes 53, 73, 113, and therefore a(n)=1.
%e A232210 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.
%e A232210 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
%t A232210 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 *)
%t A232210 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 *)
%o A232210 (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
%Y A232210 Cf. A055782, A092993, A242775, A247341, A247342, A248919.
%K A232210 nonn,base
%O A232210 1,6
%A A232210 _Vladimir Shevelev_, Sep 13 2014
%E A232210 More terms from _Peter J. C. Moses_, Sep 13 2014