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

%I A069675 #82 Jul 27 2021 15:47:23
%S A069675 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T A069675 97,101,103,107,109,307,401,409,503,509,601,607,701,709,809,907,1009,
%U A069675 2003,3001,4001,4003,4007,5003,5009,6007,7001,8009,9001,9007,10007,10009
%N A069675 Primes all of whose internal digits (if any) are 0.
%C A069675 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
%C A069675 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
%H A069675 Charles R Greathouse IV, <a href="/A069675/b069675.txt">Table of n, a(n) for n = 1..263</a>
%H A069675 Makoto Kamada, <a href="https://stdkmd.net/nrr/records.htm#k10np1">Prime numbers of the form k*10^n+1</a>
%H A069675 Seth A. Troisi, <a href="/A069675/a069675_1.png">Plot of log(log(a(n))) for 1 <= n <= 434</a>
%H A069675 Seth A. Troisi, <a href="/A069675/a069675.txt">a(n) for n = 1 .. 434, a(n) in form "a * 10 ^ d + b"</a>
%F A069675 a(n) >> 10^(n/24). - _Charles R Greathouse IV_, Sep 14 2015
%e A069675 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
%p A069675 A := {}:
%p A069675 for n to 1000 do
%p A069675   p := ithprime(n):
%p A069675   d := convert(p, base, 10):
%p A069675   s := 0:
%p A069675   for m from 2 to nops(d)-1 do
%p A069675     s := s+d[m]:
%p A069675   end do
%p A069675   if s = 0 then
%p A069675     A := `union`(A, {p})
%p A069675   end if:
%p A069675 end do:
%p A069675 A := A
%p A069675 # _César Eliud Lozada_, Sep 04 2012
%p A069675 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
%t A069675 Select[Prime[Range[1, 100000]], IntegerLength[#] < 3 || Union@Rest@Most@IntegerDigits[#, 10] == {0} &] (* _Harlan J. Brothers_, Aug 17 2015 *)
%t A069675 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 *)
%o A069675 (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
%o A069675 (Python)
%o A069675 from sympy import isprime
%o A069675 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
%Y A069675 Cf. A069676-A069684, A164968.
%K A069675 nonn,base
%O A069675 1,1
%A A069675 _Amarnath Murthy_, Apr 06 2002
%E A069675 Offset corrected and name changed by _Arkadiusz Wesolowski_, Sep 07 2011