A217063 Primes that remain prime when a single "3" digit is inserted between any two adjacent decimal digits.

11, 17, 19, 23, 29, 31, 37, 41, 43, 61, 73, 79, 89, 97, 101, 103, 127, 167, 173, 181, 211, 233, 239, 251, 271, 283, 307, 331, 359, 373, 439, 491, 509, 523, 547, 599, 673, 709, 733, 769, 877, 887, 937, 941, 991, 1033, 1229, 1381, 1619, 1721, 1759, 1789, 1901

Offset: 1

Author

Paolo P. Lava, Sep 26 2012

Keywords

• nonn
• base

Examples

			212881 is prime and also 2128831, 2128381, 2123881, 213288 and 2312881.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000 (first 275 terms from Paolo Lava)

Crossrefs

Cf. A050674, A050711-A050719, A069246, A159236, A215417, A215419-A215421, A217044-A217047, A217062-A217065.

Programs

Magma

[p: p in PrimesInInterval(11, 2000) | forall{m: t in [1..#Intseq(p)-1] | IsPrime(m) where m is (Floor(p/10^t)*10+3)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 26 2012

Maple

with(numtheory);
A217063:=proc(q,x)
local a,b,c,i,n,ok;
for n from 5 to q do
  a:=ithprime(n); b:=0; while a>0 do b:=b+1; a:=trunc(a/10); od; a:=ithprime(n); ok:=1;
    for i from 1 to b-1 do
      c:=a+9*10^i*trunc(a/10^i)+10^i*x;  if not isprime(c) then ok:=0; break; fi; od;
    if ok=1 then print(ithprime(n)); fi; od; end:
A217063(1000000,3);

PARI

is(n)=my(v=concat([""], digits(n))); for(i=2, #v-1, v[1]=Str(v[1], v[i]); v[i]=3; if(i>2, v[i-1]=""); if(!isprime(eval(concat(v))), return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 26 2012

Python

from sympy import isprime, primerange
def ok(p):
    if p < 10: return False
    s = str(p)
    return all(isprime(int(s[:i] + "3" + 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(1901)) # Michael S. Branicky, Nov 17 2021