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A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.

%I A350853 #21 Mar 11 2022 12:42:02
%S A350853 2,3,11,23,31,13,29,7,17,19,37,41,59,79,67,107,47,61,43,113,71,109,89,
%T A350853 53,157,97,83,101,73,173,131,223,149,127,197,137,373,139,167,163,179,
%U A350853 151,191,193,241,317,211,229,281,103,227,233,283,251
%N A350853 a(1) = 2, a(2) = 3; a(n) is the smallest prime not included earlier such that concatenation of three successive terms is a prime.
%C A350853 Not a permutation of the primes. 5 never appears, since numbers m mod 10 = 5 are divisible by 5, and concatenation of 2 previous terms and 5 guarantee a composite number. - _Michael De Vlieger_, Feb 16 2022
%H A350853 Haines Hoag, <a href="/A350853/b350853.txt">Table of n, a(n) for n = 1..31499</a>
%H A350853 Michael De Vlieger, <a href="/A350853/a350853.png">Scatterplot of a(n)</a> for n = 1..2^14, showing records in red and local minima (aside from q = 5, which never appears) in blue.
%e A350853 From _Michael De Vlieger_, Feb 16 2022: (Start)
%e A350853 a(3) = 11 since 235 and 237 are composite, but 2311 is prime.
%e A350853 a(4) = 23 since 3115, 3117, 31113, 31117, and 31119 are composite, but 31123 is prime.
%e A350853 a(5) = 31 since 11235, 11237, 112313, 112317, 112319, and 112329 are composite, but 112331 is prime. (End)
%t A350853 a[1]=2; a[2]=3; a[n_]:=a[n]=(k=2; While[!PrimeQ[FromDigits@Join[Flatten[IntegerDigits/@{a[n-2],a[n-1]}],IntegerDigits@k]]||MemberQ[Array[a,n-1],k],k=NextPrime@k];k);Array[a,54] (* _Giorgos Kalogeropoulos_, Jan 19 2022 *)
%Y A350853 Cf. A073641, A000040.
%K A350853 nonn,base
%O A350853 1,1
%A A350853 _Haines Hoag_, Jan 18 2022