A068679 Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

1, 3, 7, 13, 31, 49, 63, 81, 91, 99, 103, 109, 117, 123, 151, 181, 193, 213, 231, 279, 319, 367, 427, 459, 571, 601, 613, 621, 697, 721, 801, 811, 951, 987, 1113, 1117, 1131, 1261, 1821, 1831, 1939, 2101, 2149, 2211, 2517, 2611, 3151, 3219, 4011, 4411, 4519, 4887, 5031, 5361, 6231, 6487, 6871, 7011, 7209, 8671, 9141, 9801, 10051

Offset: 1

Author

Amarnath Murthy, Mar 02 2002

Keywords

• base
• nonn

Comments

If R(p) = (10^p-1)/9 is a prime then (10^(p-1)-1)/9 belongs to this sequence.

Examples

			123 belongs to this sequence as the numbers 1123, 1213, 1231 obtained by inserting a 1 in all possible ways are all primes.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..3314 (terms < 2*10^13, first 1929 terms from Chai Wah Wu)
• C. Caldwell, Prime Pages

Crossrefs

Cf. A068673, A068674, A068677, A069246.

Programs

Mathematica

d[n_]:=IntegerDigits[n]; ins[n_]:=FromDigits/@Table[Insert[d[n],1,k],{k,Length[d[n]]+1}]; Select[Range[10060],And@@PrimeQ/@ins[#] &] (* Jayanta Basu, May 20 2013 *)
Select[Range[11000],AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ #],1,n],{n,IntegerLength[#]+1}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

Python

from sympy import isprime
A068679_list, n = [], 1
while len(A068679_list) < 1000:
    if isprime(10*n+1):
        s = str(n)
        for i in range(len(s)):
            if not isprime(int(s[:i]+'1'+s[i:])):
                break
        else:
            A068679_list.append(n)
    n += 1 # Chai Wah Wu, Oct 02 2019

Extensions

More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

More terms from Vladeta Jovovic, Apr 16 2002