A360781 Primes p such that at least one number remains prime when p is bracketed by a single digit d; that is, at least one instance of d//p//d is prime where // means concatenation.

2, 3, 5, 7, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 101, 103, 107, 109, 113, 131, 139, 149, 151, 157, 163, 173, 179, 191, 193, 197, 211, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Author

Harvey P. Dale, Feb 20 2023

Keywords

• nonn
• base

Comments

The bracketing digit d must be 1, 3, 7, or 9.

Examples

			263 is included because 263 is a prime and 32633 (and also 92639) is a prime.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A059694, A069687, A069688, A069689, A069690.

Programs

Maple

q:= p-> ormap(isprime, map(d-> parse(cat(d, p, d)), [1, 3, 7, 9])):
select(q, [ithprime(i)$i=1..67])[];  # Alois P. Heinz, Feb 22 2023

Mathematica

brkQ[p_]:=AnyTrue[Table[FromDigits[Join[{d},IntegerDigits[p],{d}]],{d,{1,3,7,9}}],PrimeQ]; Select[Prime[Range[100]],brkQ]

PARI

is(p) = my(d=digits(p)); forstep(k=1, 9, 2, if (isprime(fromdigits(concat(k, concat(d,k)))), return(1)));
isok(p) = if (isprime(p), is(p)); \\ Michel Marcus, Feb 20 2023

Python

from sympy import isprime, nextprime
from itertools import islice
def agen(): # generator of terms
    p = 2
    while True:
        sp = str(p)
        if any(isprime(int(d+sp+d)) for d in "1379"):
            yield p
        p = nextprime(p)
print(list(islice(agen(), 57))) # Michael S. Branicky, Feb 20 2023

Formula

Union of A069687, A069688, A069689, A069690. - Alois P. Heinz, Feb 22 2023