A051362 Primes remaining prime if any digit is deleted (zeros allowed).

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1013, 1031, 1097, 1499, 1997, 2239, 2293, 3137, 4019, 4919, 6173, 7019, 7433, 9677, 10193, 10613, 11093, 19973, 23833, 26833, 30011, 37019, 40013, 47933, 73331, 74177

Offset: 1

Author

Harvey P. Dale, May 31 2000

Keywords

• nonn
• base
• nice

Comments

These might be called "super-prime numbers". - Jaime Gutierrez (jgutierrez(AT)matematicas.net), Sep 27 2007

A proper subset of A034895. - Robert G. Wilson v, Oct 12 2014

The largest known number in this sequence is a 274-digit prime consisting of 163 4s, followed by 80 0s, followed by 31 1s. See the CodeGolf link. - Dmitry Kamenetsky, Feb 26 2021

Links

• Giovanni Resta, Table of n, a(n) for n = 1..201 (terms < 10^13; first 100 terms from T. D. Noe)
• CodeGolf StackExchange, Find largest prime which is still a prime after digit deletion, 2013.
• Mathematics StackExchange, Deleting any digit yields a prime, 2011.
• Mathematics StackExchange, Largest prime that remains prime when any one of its digits is deleted, 2021.

Crossrefs

Cf. A034302, A010051, A000040, A034895.

Programs

Haskell

import Data.List (inits, tails)
a051362 n = a051362_list !! (n-1)
a051362_list = filter p $ drop 4 a000040_list where
   p x = all (== 1) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011, Aug 24 2011

Mathematica

rpQ[n_]:=Module[{idn=IntegerDigits[n]},And@@PrimeQ[FromDigits/@ Subsets[ IntegerDigits[ n],{Length[idn]-1}]]]; Select[Prime[Range[40000]], rpQ]
prpQ[n_]:=AllTrue[FromDigits/@Table[Delete[IntegerDigits[n],d],{d,IntegerLength[ n]}],PrimeQ]; Select[Prime[Range[7500]],prpQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 27 2020 *)

PARI

is(n)=my(v=Vec(Str(n)),k);for(i=1, #v, k=eval(concat(vecextract(v, 2^#v-1-2^(i-1))));if(!isprime(k),return(0)));isprime(n) \\ Charles R Greathouse IV, Oct 05 2011

Python

from sympy import isprime
def ok(n):
    if n < 10 or not isprime(n): return False
    s = str(n)
    return all(isprime(int(s[:i]+s[i+1:])) for i in range(len(s)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 02 2023

Sage

def is_A051362(n):
    prime = is_prime(n)
    if prime:
        L = ZZ(n).digits(10)
        for k in range(len(L)):
            K = L[:]; del K[k]
            prime = is_prime(ZZ(K, base=10))
            if not prime: break
    return prime
A051362_list = lambda n: filter(is_A051362, range(n))
A051362_list(77777) # Peter Luschny, Jul 17 2014