

A069675


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


21



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
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OFFSET

1,1


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 subset 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


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"


FORMULA

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


EXAMPLE

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


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 *)


PROG

(PARI) go(n)=my(v=List(primes(4)), t); for(d=1, n1, 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


CROSSREFS

Cf. A069676A069684, A164968.
Sequence in context: A163849 A124591 A164837 * A049585 A049549 A030291
Adjacent sequences: A069672 A069673 A069674 * A069676 A069677 A069678


KEYWORD

nonn,base


AUTHOR

Amarnath Murthy, Apr 06 2002


EXTENSIONS

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


STATUS

approved



