A382899 The smallest n-digit prime that turns composite at each step as its digits are successively appended, starting from the first.

2, 11, 101, 1013, 10007, 100003, 1000003, 10000019, 100000007, 1000000007, 10000000019, 100000000003, 1000000000061, 10000000000037, 100000000000031, 1000000000000037, 10000000000000061, 100000000000000013, 1000000000000000003, 10000000000000000051

Offset: 1

Author

Jean-Marc Rebert, Apr 08 2025

Keywords

• nonn
• base

Examples

			a(1) = 2, because 2 is prime, 22 = 2*11 is composite, while no smaller one-digit prime exhibits this property.
a(2) = 11, because 11 is prime, 111 = 3*37 and 1111 = 11*101 are composite, while no smaller two-digit prime exhibits this property.
a(4) = 1013, because 1013 is prime, 10131 = 3 * 11 * 30, 101310 = 2 * 3 * 5 * 11 * 307, 1013101 = 227 * 4463 and 10131013 = 73 * 137 * 1013 are composite, while no smaller 4-digit prime exhibits this property.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003617, A382981.

Programs

PARI

isok(p, n) = my(d=digits(p)); for (i=1, #d, p = 10*p+d[i]; if (isprime(p), return(0));); return(1);
a(n) = my(p=nextprime(10^(n-1))); while (!isok(p, n), p = nextprime(p+1)); p; \\ Michel Marcus, Apr 09 2025

Python

from sympy import isprime, nextprime
def c(s): # check if prime p's string of digits meets the concatenation condition
    return not any(isprime(int(s:=s+c)) for c in s)
def a(n):
    p = nextprime(10**(n-1))
    while not c(str(p)): p = nextprime(p)
    return p
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 09 2025