A030459 -id:A030459 - cp's OEIS frontend

A174287 Smallest natural square base q = q(k) that concatenation prime(k)//prime(k+1)//q^2 (k = 1, 2, ...) is a prime number.

3, 3, 1, 11, 1, 1, 1, 1, 1, 1, 3, 7, 31, 13, 9, 1, 1, 141, 53, 37, 9, 11, 1, 7, 61, 7, 17, 13, 17, 1, 17, 11, 7, 23, 7, 27, 27, 7, 1, 9, 19, 29, 7, 29, 19, 3, 3, 1, 43, 67, 1, 7, 7, 9, 9, 1, 13, 21, 7, 7, 7, 1, 1, 43, 1, 1, 57, 1, 67, 7, 17

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 15 2010

Note two consecutive primes prime(k)//prime(k+1)

Necessarily q is odd and has end digit 1, 3, 7 or 9

			3^2=9, 239 = prime(52) => q(1) = 3
359 = prime(72) => q(2) = 3
k=18, prime(18) = 61, 141^2 = 19881, 616719881 = prime(32151650) => q(18) = 141

J.-P. Allouche, J. Shallit: Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003

A000290, A030461, A030459, A030469, A171154, A174031, A174034

A178466 Primes prime(k) such that the concatenation prime(k+1)//prime(k) is also prime.

Original entry on oeis.org

3, 47, 53, 61, 131, 173, 199, 211, 233, 257, 353, 523, 587, 607, 619, 647, 653, 751, 797, 971, 991, 997, 1103, 1123, 1231, 1381, 1553, 1777, 1913, 1973, 1987, 2297, 2333, 2341, 2399, 2677, 2861, 3049, 3191, 3259, 3607, 3637, 3761, 3989

Offset: 1

Carmine Suriano, Jan 27 2011

53, 211, 653, 997, ... are also in A088712.
The role of the two primes is swapped in comparison to A030459.
The result of the concatenation is in A088784.

			The prime 53 is in the sequence because the next prime is 59 and 5953 is a prime.

Cf. A030459, A030460, A088712.

read("transforms") ;
for n from 1 to 600 do p := ithprime(n) ; q := nextprime(p) ; r := digcat2(q,p) ; if isprime(r) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Jan 27 2011

Transpose[Select[Partition[Prime[Range[600]],2,1],PrimeQ[FromDigits[ Flatten[ IntegerDigits/@Reverse[#]]]]&]][[1]]  (* Harvey P. Dale, Feb 02 2011 *)

a(n) = A151799(A088712(n)).

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.

Original entry on oeis.org

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

A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

Original entry on oeis.org

3, 2, 13681, 467, 127787377, 200603842261

Offset: 0

Giovanni Resta, May 14 2020

a(6) > 10^13.

			Let "|" denote concatenation.
3 | 5 = 35, which is not prime, so a(0) = 3.
2 | 3 = 23 (prime), 23 | 29 = 2329 (composite), so a(1) = 2.
13681 | 13687 (prime), 1368113687 | 1368113699 (prime), 13681136871368113699 | 13681136871368113711 (composite), so a(2) = 13681.

Carlos Rivera, Puzzle 29. P_i = P_(i-1) & nxtprm(P_(i-1)), P_i = prime for i => 1, The Prime Puzzles and Problems Connection.

Cf. A030459, A034593, A036339, A036340, A036341.

a[n_] := Block[{pp=1, p, q, c=-1}, While[ c!=n, c=0; p = pp = NextPrime@ pp; While[ PrimeQ[ q = FromDigits[ Join @@ IntegerDigits@{p, NextPrime@ p}]], c++; p = q]]; pp]; a /@ Range[0, 3]

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.

cp's OEIS Frontend

A174287 Smallest natural square base q = q(k) that concatenation prime(k)//prime(k+1)//q^2 (k = 1, 2, ...) is a prime number.

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A178466 Primes prime(k) such that the concatenation prime(k+1)//prime(k) is also prime.

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A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

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

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