A069675 Primes all of whose internal digits (if any) are 0.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 1009, 2003, 3001, 4001, 4003, 4007, 5003, 5009, 6007, 7001, 8009, 9001, 9007, 10007, 10009

Offset: 1

Author

Amarnath Murthy, Apr 06 2002

Keywords

• nonn
• base

Comments

Despite their initial density, these primes are rare. The value of a(310) = 9*10^2914 + 7. Beginning with a(54), this is a subsequence of A164968. Indeed, these could be called the "naughtiest" primes. - Harlan J. Brothers, Aug 17 2015

There are expected to be infinitely many terms, but growing very rapidly, something like a(n) ~ exp(exp(const * n)). - Robert Israel, Aug 17 2015

Examples

			4001 is in the sequence because it is prime and all the internal digits (the digits between 4 and 1) are zero. - _Michael B. Porter_, Aug 11 2016
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..263
• Makoto Kamada, Prime numbers of the form k*10^n+1
• Seth A. Troisi, Plot of log(log(a(n))) for 1 <= n <= 434
• Seth A. Troisi, a(n) for n = 1 .. 434, a(n) in form "a * 10 ^ d + b"

Crossrefs

Cf. A069676-A069684, A164968.

Programs

Maple

A := {}:
for n to 1000 do
  p := ithprime(n):
  d := convert(p, base, 10):
  s := 0:
  for m from 2 to nops(d)-1 do
    s := s+d[m]:
  end do
  if s = 0 then
    A := `union`(A, {p})
  end if:
end do:
A := A
# César Eliud Lozada, Sep 04 2012
select(isprime, [$1..9, seq(seq(seq(10^d*a+b, b=1..9),a=1..9), d=1..10)]); # Robert Israel, Aug 18 2015

Mathematica

Select[Prime[Range[1, 100000]], IntegerLength[#] < 3 || Union@Rest@Most@IntegerDigits[#, 10] == {0} &] (* Harlan J. Brothers, Aug 17 2015 *)
Select[Join[Range[1, 99], Flatten[Table[a*10^d + b, {d, 2, 50}, {a, 1, 9}, {b, 1, 9}]]], PrimeQ[#] &] (* Seth A. Troisi, Aug 03 2016 *)

PARI

go(n)=my(v=List(primes(4)),t); for(d=1,n-1, for(i=1,9, forstep(j=1,9,[2,4,2], if(isprime(t=10^d*i+j), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Sep 14 2015

Python

from sympy import isprime
print([2, 3, 5, 7] + list(filter(isprime, (a*10**d+b for d in range(1, 101) for a in range(1, 10) for b in [1, 3, 7, 9])))) # Michael S. Branicky, May 08 2021

Formula

a(n) >> 10^(n/24). - Charles R Greathouse IV, Sep 14 2015

Extensions

Offset corrected and name changed by Arkadiusz Wesolowski, Sep 07 2011