A034302 Zeroless primes that remain prime if any digit is deleted.

23, 37, 53, 73, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1499, 1997, 2239, 2293, 3137, 4919, 6173, 7433, 9677, 19973, 23833, 26833, 47933, 73331, 74177, 91733, 93491, 94397, 111731, 166931, 333911, 355933, 477797, 477977

Offset: 1

Author

David W. Wilson

Keywords

• base
• nonn
• nice

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..118 (terms 1..79 from T. D. Noe, terms 80..103 from Charles R Greathouse IV)
• StackExchange, Deleting any digit yields a prime

Crossrefs

Cf. A034303, A034304, A034305, A051362, A010051, A038618.

Programs

Haskell

import Data.List (inits, tails)
a034302 n = a034302_list !! (n-1)
a034302_list = filter f $ drop 4 a038618_list where
   f x = all (== 1) $ map (a010051 . read) $
             zipWith (++) (inits $ show x) (tail $ tails $ show x)
-- Reinhard Zumkeller, Dec 17 2011

Mathematica

rpnzQ[n_]:=Module[{idn=IntegerDigits[n]},Count[idn,0]==0 && And@@ PrimeQ[FromDigits/@ Subsets[IntegerDigits[n], {Length[idn]-1}]]]; Select[Prime[Range[40000]],rpnzQ]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

is(n)=my(d=digits(n),t=2^#d-1); if(vecmin(d)==0, return(0)); for(i=0,#d-1, if(!isprime(fromdigits(vecextract(d,t-2^i))), return(0))); isprime(n) \\ Charles R Greathouse IV, Jun 23 2017

Python

from itertools import product
from sympy import isprime
A034302_list, m = [23, 37, 53, 73], 7
for l in range(1,m-1): # generate all terms less than 10^m
    for d in product('123456789',repeat=l):
        for e in product('1379',repeat=2):
            s = ''.join(d+e)
            if isprime(int(s)):
                for i in range(len(s)):
                    if not isprime(int(s[:i]+s[i+1:])):
                        break
                else:
                    A034302_list.append(int(s)) # Chai Wah Wu, Apr 05 2021