A068695 Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

1, 3, 1, 1, 3, 1, 1, 3, 7, 1, 3, 7, 1, 9, 1, 3, 3, 1, 1, 11, 1, 3, 3, 1, 1, 3, 1, 1, 3, 7, 1, 17, 1, 7, 3, 7, 3, 3, 7, 1, 9, 1, 1, 3, 7, 1, 9, 7, 1, 3, 13, 1, 23, 1, 7, 3, 1, 7, 3, 1, 3, 11, 1, 1, 3, 1, 3, 3, 1, 1, 9, 7, 3, 3, 1, 1, 3, 7, 7, 9, 1, 1, 9, 19, 3, 3, 7, 1, 23, 7, 1, 9, 7, 1, 3, 7, 1, 3, 1, 9, 3, 1

Offset: 1

Author

Amarnath Murthy, Mar 03 2002

Keywords

• base
• easy
• nonn

Comments

Max Alekseyev (see link) shows that a(n) always exists. Note that although his argument makes use of some potentially large constants (see the comments in A060199), the proof shows that a(n) exists for all n. - N. J. A. Sloane, Nov 13 2020

Many numbers become prime by appending a one-digit odd number. Some numbers (such as 20, 32, 51, etc.) require a 2-digit odd number (A032352 has these). In the first 100000 values of n there are only 22 that require a 3-digit odd number (A091089). There probably are some values that require odd numbers of 4 or more digits, but these are likely to be very large. - Chuck Seggelin, Dec 18 2003

Examples

			a(20)=11 because 11 is the minimum odd number which when appended to 20 forms a prime (201, 203, 205, 207, 209 are all nonprime, 2011 is prime).
		

Links

• David W. Wilson, Table of n, a(n) for n=1..10000
• Max Alekseyev, Given n, there is a k such that the concatenation n||k is a prime, Nov 09 2020
• Index entries for primes involving decimal expansion of n

Crossrefs

Cf. A032352 (a(n) requires at least a 2 digit odd number), A091089 (a(n) requires at least a 3 digit odd number).

Cf. also A060199, A228325, A336893.

Programs

Mathematica

d[n_]:=IntegerDigits[n]; t={}; Do[k=1; While[!PrimeQ[FromDigits[Join[d[n],d[k]]]],k++]; AppendTo[t,k],{n,102}]; t (* Jayanta Basu, May 21 2013 *)
mon[n_]:=Module[{k=1},While[!PrimeQ[n*10^IntegerLength[k]+k],k+=2];k]; Array[mon,110] (* Harvey P. Dale, Aug 13 2018 *)

PARI

A068695=n->for(i=1,oo,ispseudoprime(eval(Str(n,i)))&&return(i)) \\ M. F. Hasler, Oct 29 2013

Python

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(1) if isprime(int(str(n)+str(k))))
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 18 2022

Extensions

More terms from Chuck Seggelin, Dec 18 2003

Entry revised by N. J. A. Sloane, Feb 20 2006

More terms from David Wasserman, Feb 14 2006