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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 (list; graph; refs; listen; history; text; internal format)
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

Robert Israel, Plot of log(log(a(n))) for 1 <= n <= 293

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, 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

CROSSREFS

Cf. A069676-A069684, 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

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Last modified August 20 13:04 EDT 2017. Contains 290835 sequences.