A309191 -id:A309191 - cp's OEIS frontend

A333935 Take a prime p and concatenate it with the next prime after p, q. If the result p||q is prime then take p+q and concatenate the sum to the next prime after q, r. If the result (p+q)||r is prime then take p+q+r and concatenate it to the next prime (s) after r, getting (p+r+r)||s, and so on. a(n) is the least prime for which this process continues for n steps.

2, 31, 331, 832757, 2683591, 6363925717, 1478441195963, 8996779470869

Offset: 1

Michel Marcus, Apr 11 2020

i||j denotes the concatenation of the decimal expansions of i and j.

			For p = prime(1) = 2 and prime(2) = 3 the concatenation 2||3 is prime; then 2+3 = 5 and the concatenation of 5 and prime(3) = 5 is 5||5 whicch is not a prime; so the process works for just one step, and a(1) = 2
For p = prime(11)= 31 and prime(12) = 37, the concatenation 31||37 is prime; then 31+37 = 68 and the concatenation of 68 and prime(13) = 41 is 68||41 which is prime; then 68+41 = 109 and the concatenation of 109 and prime(14) = 43 is 109||43 = 31*353 which is not prime; so the process ends after two steps. There is no prime < 31 such that the process continues for 2 steps, so a(2) = 31.

Carlos Rivera, Puzzle 995. Another sequence of primes, The Prime Puzzles and Problems Connection.

Cf. A030459, A309191.

catQ[{m_, n_}] := PrimeQ @ FromDigits @ Join[IntegerDigits[m], IntegerDigits[n]]; f[{c_, p_}] := {c + p, NextPrime[p]}; s[p_] := Length @ NestWhileList[f, {0, p}, catQ] - 2; a[n_] := Module[{p = 2}, While[s[p] != n, p = NextPrime[p]]; p]; Array[a, 5] (* Amiram Eldar, Apr 15 2020 *)

isok(p, n) = {my(s = p); for (i=1, n, my(q = nextprime(p+1)); if (!isprime(eval(Str(s, q))), return (0)); s += q; p = q;); return(1);}
a(n) = my(p=prime(1)); while (!isok(p, n), p=nextprime(p+1)); p;

a(6) from Emmanuel Vantieghem, Apr 15 2020

a(7)-a(8) from Giovanni Resta, Apr 15 2020

A359406 Integers k such that the concatenation of k consecutive primes starting at 31 is prime.

Original entry on oeis.org

1, 2, 3, 23, 43, 141

Offset: 1

Mikk Heidemaa, Dec 30 2022

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.
a(7): 3472 corresponds to a 15968-digit probable prime (certification in progress).
For a(8), k > 15000 (if it exists).
a(8) > 30000. - Tyler Busby, Feb 13 2023

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

Cf. A069151, A309191, A030996, A280894.

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

A380092 Number of consecutive primes after prime(n) before their concatenation fails to produce a prime.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0

Offset: 1

Robert G. Wilson v, Jan 12 2025

			a(1) = 1 since prime(1) = 2 can be concatenated with the next prime 3 to 23 which is prime, but the next concatenation with 5 is 235 which is not prime.
a(2) = 0 since prime(2) = 3 but concatenating the next prime 5 is 35 which is not prime.
a(11) = 2 since prime(11) = 31 concatenates: 3137 is prime, 313741 is prime, but 31374143 is not prime.

Cf. A000040, A030459, A219271, A309191.

a[n_] := Block[{k = 0, p = Prime@ n}, While[ PrimeQ[ FromDigits[ Flatten[ IntegerDigits[ NextPrime[p, Range[0, k]]]]]], k++]; --k]; Array[a, 105]

a(n) = 0 iff (p_(n+1) - p_n)/2 == 1 (mod 2).

a(n) > 0 iff n is in A030459.

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A359406 Integers k such that the concatenation of k consecutive primes starting at 31 is prime.

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A380092 Number of consecutive primes after prime(n) before their concatenation fails to produce a prime.

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