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A069675
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Primes all of whose internal digits (if any) are 0.
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21
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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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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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
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MAPLE
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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
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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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