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Integers k such that the concatenation of k consecutive primes starting at 31 is prime.
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%I #58 Feb 14 2023 02:35:43

%S 1,2,3,23,43,141

%N Integers k such that the concatenation of k consecutive primes starting at 31 is prime.

%C The corresponding primes (p) known (31, 3137, 313741, ...) have an even number of digits and p (mod 10) == 1|7. For those at a(1)...a(6), p (mod 3) == p (mod 5) holds.

%C a(7): 3472 corresponds to a 15968-digit probable prime (certification in progress).

%C For a(8), k > 15000 (if it exists).

%C a(8) > 30000. - _Tyler Busby_, Feb 13 2023

%e 2 is a term because the consecutive primes 31 and 37 concatenated to 3137 yield another prime.

%t UpToK[k_] := Block[{a := FromDigits @ Flatten @ IntegerDigits @ Join[{}, Prime @ Range[11, i]]}, Reap[ Do[ If[ PrimeQ[a], Sow[i - 10], Sow[Nothing]], {i, k}]]][[2, 1]]; UpToK[3500] (* or *)

%t UpToK[k_] := Flatten @ Parallelize @ MapIndexed[ If[ PrimeQ[#1], #2, Nothing] &, DeleteCases[ FromDigits /@ Flatten /@ IntegerDigits @ Prime @ Range[11, Range[k]], 0]]; UpToK[3500]

%Y Cf. A069151, A309191, A030996, A280894.

%K nonn,base,more

%O 1,2

%A _Mikk Heidemaa_, Dec 30 2022