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

1,2

Comments

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A068673, A068674, A068677, A069246.

Sequence in context: A309738 A161218 A351318 * A006978 A060424 A119962

Adjacent sequences: A068676 A068677 A068678 * A068680 A068681 A068682

Keyword

base,nonn

Author

Amarnath Murthy, Mar 02 2002

Extensions

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

More terms from Vladeta Jovovic, Apr 16 2002

Status

approved