A217047 Primes that remain prime when a single "8" digit is inserted between any two adjacent digits.

11, 23, 47, 83, 131, 173, 179, 233, 353, 389, 521, 569, 641, 683, 839, 887, 911, 971, 983, 1229, 1289, 1913, 2087, 2663, 2837, 2879, 3329, 3671, 3677, 3803, 3821, 4259, 4409, 4817, 4871, 4889, 5237, 5477, 5693, 6449, 6581, 6863, 7283, 7487, 7583, 7823, 7853

Offset: 1

Author

Paolo P. Lava, Sep 25 2012

Keywords

• nonn
• base

Comments

These numbers are either isolated primes or the smaller of a pair of twin primes. - Davide Rotondo, Mar 11 2025

Examples

			325421 is prime and also 3254281, 3254821, 3258421, 3285421 and 3825421.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217046, A217062-A217065.

Programs

Magma

[p: p in PrimesInInterval(11,8000) | forall{m: t in [1..#Intseq(p)-1] | IsPrime(m) where m is (Floor(p/10^t)*10+8)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 26 2012

Maple

A217044:=proc(q,x) local a,b,c,d,i,k,n,ok,v; v:=[]; a:=10;
for n from 1 to q do a:=nextprime(a); d:=length(a); ok:=1;
for k from 1 to d-1 do b:=a mod 10^k; c:=trunc(a/10^k); i:=x*10^k+b; i:=c*10^length(i)+i;
if not isprime(i) then ok:=0; break; fi; od; if ok=1 then v:=[op(v),a]; fi; od; op(v); end:
A217044(10^3,8);

PARI

is(n)=my(v=concat([""],digits(n)));for(i=2,#v-1,v[1]=Str(v[1], v[i]); v[i]=8;if(i>2,v[i-1]="");if(!isprime(eval(concat(v))), return(0)));isprime(n) \\ Charles R Greathouse IV, Sep 25 2012

Python

from sympy import isprime, primerange
def ok(p):
    if p < 10: return False
    s = str(p)
    return all(isprime(int(s[:i] + "8" + s[i:])) for i in range(1, len(s)))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(7854)) # Michael S. Branicky, Nov 23 2021