A034305 Zeroless nonprimes that remain nonprime if any digit is deleted.

14, 16, 18, 44, 46, 48, 49, 64, 66, 68, 69, 81, 84, 86, 88, 91, 94, 96, 98, 99, 122, 124, 125, 126, 128, 142, 144, 145, 146, 148, 152, 154, 155, 156, 158, 162, 164, 165, 166, 168, 182, 184, 185, 186, 188, 212, 214, 215, 216, 218, 221, 222, 224, 225, 226, 228

Offset: 1

Author

David W. Wilson

Keywords

• base
• nonn
• nice

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Crossrefs

Subsequence of A052382.

Cf. A034302, A034303, A034304, A010051.

Programs

Haskell

a034305 n = a034305_list !! (n-1)
a034305_list = filter f $ drop 9 a052382_list where
  f x = a010051' x == 0 &&
        (all (== 0) $ map (a010051' . read) $
         zipWith (++) (inits $ show x) (tail $ tails $ show x))
-- Reinhard Zumkeller, May 10 2015

Mathematica

npQ[n_]:=!PrimeQ[n]&&FreeQ[IntegerDigits[n],0]&&AllTrue[FromDigits/@ Table[Drop[IntegerDigits[n],{k}],{k,IntegerLength[n]}],!PrimeQ[#]&]; Select[Range[10,300],npQ](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 19 2021 *)

PARI

is(n)=my(d=digits(n)); if(#d<2 || vecmin(d)<1 || isprime(n), return(0)); for(i=0,#d-1, if(isprime(fromdigits(vecextract(d,2^#d-1-2^i))), return(0))); 1 \\ Charles R Greathouse IV, Jun 25 2017

Python

from sympy import isprime
def ok(n):
    if n < 10 or isprime(n): return False
    s = str(n)
    return "0" not in s and not any(isprime(int(s[:i]+s[i+1:])) for i in range(len(s)))
print([k for k in range(229) if ok(k)]) # Michael S. Branicky, Jan 15 2023

Extensions

Definition corrected by T. D. Noe, Apr 02 2008

Single-digit terms removed again by Georg Fischer, Jun 21 2021