A018800 Smallest prime that begins with n.

11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 211, 223, 23, 241, 251, 263, 271, 281, 29, 307, 31, 3203, 331, 347, 353, 367, 37, 383, 397, 401, 41, 421, 43, 443, 457, 461, 47, 487, 491, 503, 5101, 521, 53, 541, 557, 563, 571, 587, 59

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base

Comments

Conjecture: If a(n) = (n concatenated with k) then k < n. - Amarnath Murthy, May 01 2002

a(n) always exists. Proof. Suppose n is L digits long, and consider the numbers between n*10^B and n*10^B+10^C, where B > C are both large compared with L. All such numbers begin with the digits of n. Using the upper and lower bounds on pi(x) from Theorem 1 of Rosser and Schoenfeld, it follows that for sufficiently large B and C, at least one of these numbers is a prime. QED - N. J. A. Sloane, Nov 14 2014

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Paolo P. Lava)
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), pp. 64-94.
• Index entries for primes involving decimal expansion of n

Crossrefs

Cf. A030665, A068164, A068695, A062584, A088781, A085608.

A164022 is the base-2 analog.

Cf. also A258337.

Row n=1 of A262369.

Programs

Haskell

import Data.List (isPrefixOf, find); import Data.Maybe (fromJust)
a018800 n = read $ fromJust $
            find (show n `isPrefixOf`) $ map show a000040_list :: Int
-- Reinhard Zumkeller, Jul 01 2015

Maple

f:= proc(n) local x0, d,r,y;
   if isprime(n) then return(n) fi;
   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

Table[Function[d, FromDigits@ SelectFirst[ IntegerDigits@ Prime@ Range[10^4], Length@ # >= Length@ d && Take[#, Length@ d] == d &]][ IntegerDigits@ n], {n, 59}] (* Michael De Vlieger, May 24 2016, Version 10 *)

PARI

a(n{,base=10}) = for (l=0, oo, forprime (p=n*base^l, (n+1)*base^l-1, return (p))) \\ Rémy Sigrist, Jun 11 2017

Python

from sympy import isprime
def a(n):
    if isprime(n): return n
    pow10 = 10
    while True:
        t, maxt = n * pow10 + 1, (n+1) * pow10
        while t < maxt:
            if isprime(t): return t
            t += 2
        pow10 *= 10
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Nov 02 2021

Formula

a(n) = prime(A085608(n)). - Michel Marcus, Oct 19 2013