A372207 a(n) is the smallest prime number that is obtained by concatenating 2 consecutive n-digit integers.

23, 1213, 102101, 10021001, 1000810007, 100010100011, 10000241000023, 1000004210000041, 100000002100000003, 10000000081000000009, 1000000000810000000007, 100000000002100000000001, 10000000000021000000000001, 1000000000001010000000000011, 100000000000002100000000000003

Offset: 1

Author

Gonzalo Martínez, May 19 2024

Keywords

• nonn
• base

Comments

It is possible to form primes by concatenating an integer with its successor (A030458), as well as by concatenating an integer with its predecessor (A052089).

Examples

			a(3) = 102101, since when concatenating 3-digit numbers, the first three numbers obtained, in increasing order, are 100101, 101100, 101102, which are composite, while the next number in the list is 102101, which is prime.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A030458, A052089.

Programs

Python

from sympy import isprime
def a(n):
    if n == 1: return 23
    lb, ub = 10**(n-1), 10**n
    for k in range(lb, ub, 2):
        base = k*ub
        for inc in [k-1, k+1]:
            if inc >= lb and isprime(t:=base+inc):
                return t
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, May 19 2024

Extensions

a(12) and beyond from Michael S. Branicky, May 19 2024