OFFSET
1,1
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.
FORMULA
a(n) = prime(A085608(n)). - Michel Marcus, Oct 19 2013
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
STATUS
approved