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A069675 Primes with either no internal digits or all internal digits are 0. 21

%I

%S 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 97,101,103,107,109,307,401,409,503,509,601,607,701,709,809,907,1009,

%U 2003,3001,4001,4003,4007,5003,5009,6007,7001,8009,9001,9007,10007,10009

%N Primes with either no internal digits or all internal digits are 0.

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

%C 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 Charles R Greathouse IV, <a href="/A069675/b069675.txt">Table of n, a(n) for n = 1..263</a>

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/records.htm#k10np1">Prime numbers of the form k*10^n+1</a>

%H Seth A. Troisi, <a href="/A069675/a069675_1.png">Plot of log(log(a(n))) for 1 <= n <= 434</a>

%H Seth A. Troisi, <a href="/A069675/a069675.txt">a(n) for n = 1 .. 434, a(n) in form "a * 10 ^ d + b"</a>

%F a(n) >> 10^(n/24). - _Charles R Greathouse IV_, Sep 14 2015

%e 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 A := {}:

%p for n to 1000 do

%p p := ithprime(n):

%p d := convert(p, base, 10):

%p s := 0:

%p for m from 2 to nops(d)-1 do

%p s := s+d[m]:

%p end do

%p if s = 0 then

%p A := `union`(A, {p})

%p end if:

%p end do:

%p A := A

%p # _C├ęsar Eliud Lozada_, Sep 04 2012

%p 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 Select[Prime[Range[1, 100000]], IntegerLength[#] < 3 || Union@Rest@Most@IntegerDigits[#, 10] == {0} &] (* _Harlan J. Brothers_, Aug 17 2015 *)

%t 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 (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

%Y Cf. A069676-A069684, A164968.

%K nonn,base

%O 1,1

%A _Amarnath Murthy_, Apr 06 2002

%E Offset corrected and name changed by _Arkadiusz Wesolowski_, Sep 07 2011

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Last modified August 7 08:08 EDT 2020. Contains 336274 sequences. (Running on oeis4.)