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
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, 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
KEYWORD
nonn,base
AUTHOR
Amarnath Murthy, Apr 06 2002
EXTENSIONS
Offset corrected and name changed by Arkadiusz Wesolowski, Sep 07 2011
STATUS
approved