A030665 Smallest nontrivial extension of n which is prime.

11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2003, 211, 223, 233, 241, 251, 263, 271, 281, 293, 307, 311, 3203, 331, 347, 353, 367, 373, 383, 397, 401, 419, 421, 431, 443, 457, 461, 479, 487, 491, 503, 5101

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base
• nice

Comments

The argument in A069695 shows that a(n) always exists. - N. J. A. Sloane, Nov 11 2020

Examples

			For n = 1, we could append 1, 3, 7, 9, 01, etc., to make a prime, but 1 gives the smallest of these, 11, so a(1) = 11.
For n = 2, although 2 is already prime, the definition requires an appending at least one digit. 1 doesn't work because 21 = 3 * 7, but 3 does because 23 is prime. Hence a(2) = 23.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 [T. D. Noe computed the first 1000 terms]
• Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.
• Index entries for primes involving decimal expansion of n

Crossrefs

Cf. A018800, A068695, A077501.

Programs

Maple

f:= proc(n) local x0, d, r, y;
   for d from 1 do
     x0:= n*10^d;
     for r from 1 to 10^d-1 by 2 do
       if isprime(x0+r) then
          return(x0+r)
       fi
     od
   od
end proc:
seq(f(n), n=1..100); # Robert Israel, Dec 23 2014

Mathematica

A030665[n_] := Module[{d = 10, nd = 10 * n}, While[True, x = NextPrime[nd]; If[x < nd + d, Return[x]]; d *= 10; nd *= 10]]; Array[A030665, 100] (* Jean-François Alcover, Oct 19 2016, translated from Chai Wah Wu's Python code *)

PARI

apply( {A030665(n)=my(p,L); until(n+10^L++>p=nextprime(n), n*=10);p}, [1..55]) \\ M. F. Hasler, Jan 27 2025

Python

from sympy import nextprime
def A030665(n):
    d, nd = 10, 10*n
    while True:
        x = nextprime(nd)
        if x < nd+d:
            return int(x)
        d *= 10
        nd *= 10 # Chai Wah Wu, May 24 2016

Extensions

Corrected by Ray Chandler, Aug 11 2003