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A018800
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Smallest prime that begins with n.
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16
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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
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history;
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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Paolo P. Lava and 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
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FORMULA
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a(n) = prime(A085608(n)). - Michel Marcus, Oct 19 2013
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A030665, A068164, A068695, A062584, A088781, A085608.
A164022 is the base-2 analog.
Cf. also A258337.
Row n=1 of A262369.
Sequence in context: A077549 A089356 A113616 * A258337 A089566 A010191
Adjacent sequences: A018797 A018798 A018799 * A018801 A018802 A018803
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KEYWORD
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nonn,base
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AUTHOR
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David W. Wilson
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STATUS
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approved
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