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A228323 a(1)=1; thereafter a(n) is the smallest number m not yet in the sequence such that at least one of the concatenations a(n-1)||m or m||a(n-1) is prime.

%I A228323 #39 Dec 23 2024 14:53:43
%S A228323 1,3,2,9,5,21,4,7,6,13,10,19,16,27,8,11,15,23,12,17,20,29,14,33,26,47,
%T A228323 18,31,25,39,22,37,24,41,30,49,34,57,28,43,36,59,32,51,38,53,42,61,45,
%U A228323 67,58,69,55,63,44,81,35,71,48,77,50,87,62,99,40,73,46
%N A228323 a(1)=1; thereafter a(n) is the smallest number m not yet in the sequence such that at least one of the concatenations a(n-1)||m or m||a(n-1) is prime.
%C A228323 Does every number appear in the sequence?
%C A228323 If a(n) is coprime to 10, then a(n+1) exists by Dirichlet's theorem. - _Eric M. Schmidt_, Aug 20 2013 [In more detail: let a(n) have d digits, and consider the arithmetic progression k*10^d + a(n), and apply Dirichlet's theorem. This gives a number k such that the concatenation k||a(n) is prime. _N. J. A. Sloane_, Nov 08 2020]
%C A228323 The argument in A068695 shows that a(n) always exists. - _N. J. A. Sloane_, Nov 11 2020
%H A228323 Alois P. Heinz, <a href="/A228323/b228323.txt">Table of n, a(n) for n = 1..10000</a>
%H A228323 Eric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-August/011576.html">Primes by concatenation</a>, Posting to the Sequence Fans Mailing List, Aug 14 2013.
%H A228323 Michael De Vlieger, <a href="/A228323/a228323.png">Labeled log-log scatterplot of a(n)</a> n = 1..2^14, showing m coprime to 10 in red, otherwise dark blue.
%H A228323 <a href="/index/Pri#piden">Index entries for primes involving decimal expansion of n</a>
%t A228323 f[s_] := Block[{k = 2, idj = IntegerDigits@ s[[-1]]}, While[idk = IntegerDigits@ k; MemberQ[s, k] || ( !PrimeQ@ FromDigits@ Join[idj, idk] && !PrimeQ@ FromDigits@ Join[idk, idj]), k++]; Append[s, k]]; Nest[f, {1}, 66] (* _Robert G. Wilson v_, Aug 20 2013 *)
%o A228323 (Python)
%o A228323 from sympy import isprime
%o A228323 from itertools import islice
%o A228323 def c(s, t): return isprime(int(s+t)) or isprime(int(t+s))
%o A228323 def agen():
%o A228323     aset, k, mink = set(), 1, 2
%o A228323     while True:
%o A228323         an = k; aset.add(an); yield an; s, k = str(an), mink
%o A228323         while k in aset or not c(s, str(k)): k += 1
%o A228323         while mink in aset: mink += 1
%o A228323 print(list(islice(agen(), 56))) # _Michael S. Branicky_, Oct 17 2022
%Y A228323 See A228324 for the primes that arise.
%Y A228323 Cf. A069695, A228325.
%K A228323 nonn,base
%O A228323 1,2
%A A228323 _N. J. A. Sloane_, Aug 20 2013
%E A228323 More terms from _Alois P. Heinz_, Aug 20 2013