A326398 a(n) is the smallest k > 0 such that the concatenation prime(n)k is composite.

1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1

Offset: 1

Author

David James Sycamore, Jun 07 2020

Keywords

• nonn
• base

Comments

a(n) can only be 1 or 2 for any n >= 1. It appears that there are many more primes for which k = 1 than those for which k = 2 (~ 91% of the first 10^7 primes have k = 1).

Examples

			a(1) = 1 because 21 is prime, a(2) = 2 because 31 is prime (as 3 is in A023237), and 32 is composite
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A023237.

Programs

Maple

P := proc (n)
if isprime(10*ithprime(n)+1) then return 2 else 1;
end if:
end proc;
P(50);
seq(P(k), k = 1 .. 50);

Mathematica

Table[k = 1; p = Prime[n]; While[PrimeQ[10^IntegerLength[k]*p + k], k++]; k, {n, 120}] (* Michael De Vlieger, Jun 30 2025 *)

PARI

a(n) = my(k=1, p=Str(prime(n))); while (isprime(eval(concat(p, Str(k)))), k++); k; \\ Michel Marcus, Jun 07 2020

Formula

a(n) = 2 if prime(n) is in A023237, else a(n) = 1. [corrected by Chai Wah Wu, Jul 06 2020]