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A069675 Primes all of whose internal digits (if any) 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 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
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..263
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
(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
CROSSREFS
Sequence in context: A163849 A124591 A164837 * A049585 A049549 A030291
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 April 18 18:58 EDT 2024. Contains 371781 sequences. (Running on oeis4.)