A068679 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

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

(list; graph; refs; listen; history; text; internal format)

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)