A308438 a(n) is the smallest prime p whose decimal expansion begins with n and is such that the next prime is p+2n, or -1 if no such prime exists.

11, 223, 31, 401, 547, 619, 773, 8581, 9109, 10223, 1129, 12073, 130553, 14563, 150011, 161471, 17257, 18803, 191189, 20809, 210557, 225383, 237091, 240209, 2509433, 2613397, 277429, 283211, 2901649, 308153, 313409, 3204139, 3300613, 3419063, 3507739, 360091, 3727313, 3806347, 3930061, 4045421, 41018911

Offset: 1

Author

J. M. Bergot and Robert Israel, May 30 2019

Keywords

• nonn
• base
• look

Examples

			For n = 5, 547 is a prime starting with 5, and the next prime after 547 is 557 = 547 + 2*5.  Since this is the least number with these properties, a(5) = 547.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A018800, A030665.

Programs

Maple

f:= proc(n) local d,p,q;
  for d from 0 do
    p:= nextprime(n*10^d-1);
    do
      q:= nextprime(p);
      if q - p = 2*n then return p fi;
      if q >= (n+1)*10^d then break fi;
      p:= q;
    od;
  od;
end proc:
map(f, [$1..50]);

Python

from sympy import nextprime
def A308438(n):
    l, p = 1, nextprime(n)
    while True:
        q = nextprime(p)
        if q-p == 2*n:
            return p
        p = q
        if p >= (n+1)*l:
            l *= 10
            p = nextprime(n*l) # Chai Wah Wu, May 31 2019