A217040 Bases b in which the increasing concatenation of all primes smaller than b forms a prime number.

3, 4, 5, 9, 10, 15, 244, 676, 14870, 23526, 35732, 47133, 66878

Offset: 1

Author

James G. Merickel, Sep 25 2012

Keywords

• nonn
• base
• hard
• more

Comments

This sequence is a list of those bases that give prime values analogous to the prime 2357 in base 10.

Heuristically, this sequence should be infinite with approximately logarithmic density. - Charles R Greathouse IV, Sep 27 2012

Examples

			2 is the only prime less than 3, and the improper 'concatenation' of this one term is prime, so 3 is in this sequence.
In base 4, the number represented as 23 is 2*4 + 3 = 11, a prime (so 4 is included in the list); the base-5 case, similarly, yields the prime 13, as represented in base 10; 6 is not on the list because 2*6^2+3*6+5=95 is composite; and so on.
		

Crossrefs

Cf. A019518, A046035.

Programs

PARI

is(n)=isprime(subst(Pol(primes(primepi(n-1))),'x,n)) \\ Charles R Greathouse IV, Sep 26 2012

Python

from sympy import primerange, isprime
def fromdigits(d, b):
  n = 0
  for di in d: n *= b; n += di
  return n
def ok(b): return isprime(fromdigits([p for p in primerange(1, b)], b))
print([b for b in range(3, 700) if ok(b)]) # Michael S. Branicky, Mar 04 2021

Extensions

a(10) from Charles R Greathouse IV, Sep 27 2012

a(11)-a(12) from Michael S. Branicky, Jul 27 2023

a(13) from Michael S. Branicky, Aug 03 2023