cp's OEIS Frontend

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

%I A217040 #37 Aug 03 2023 10:45:46
%S A217040 3,4,5,9,10,15,244,676,14870,23526,35732,47133,66878
%N A217040 Bases b in which the increasing concatenation of all primes smaller than b forms a prime number.
%C A217040 This sequence is a list of those bases that give prime values analogous to the prime 2357 in base 10.
%C A217040 Heuristically, this sequence should be infinite with approximately logarithmic density. - _Charles R Greathouse IV_, Sep 27 2012
%e A217040 2 is the only prime less than 3, and the improper 'concatenation' of this one term is prime, so 3 is in this sequence.
%e A217040 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.
%o A217040 (PARI) is(n)=isprime(subst(Pol(primes(primepi(n-1))),'x,n)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A217040 (Python)
%o A217040 from sympy import primerange, isprime
%o A217040 def fromdigits(d, b):
%o A217040   n = 0
%o A217040   for di in d: n *= b; n += di
%o A217040   return n
%o A217040 def ok(b): return isprime(fromdigits([p for p in primerange(1, b)], b))
%o A217040 print([b for b in range(3, 700) if ok(b)]) # _Michael S. Branicky_, Mar 04 2021
%Y A217040 Cf. A019518, A046035.
%K A217040 nonn,base,hard,more
%O A217040 1,1
%A A217040 _James G. Merickel_, Sep 25 2012
%E A217040 a(10) from _Charles R Greathouse IV_, Sep 27 2012
%E A217040 a(11)-a(12) from _Michael S. Branicky_, Jul 27 2023
%E A217040 a(13) from _Michael S. Branicky_, Aug 03 2023