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Bases b in which the increasing concatenation of all primes smaller than b forms a prime number.
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%I #37 Aug 03 2023 10:45:46

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

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

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

%C Heuristically, this sequence should be infinite with approximately logarithmic density. - _Charles R Greathouse IV_, Sep 27 2012

%e 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 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 (PARI) is(n)=isprime(subst(Pol(primes(primepi(n-1))),'x,n)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Python)

%o from sympy import primerange, isprime

%o def fromdigits(d, b):

%o n = 0

%o for di in d: n *= b; n += di

%o return n

%o def ok(b): return isprime(fromdigits([p for p in primerange(1, b)], b))

%o print([b for b in range(3, 700) if ok(b)]) # _Michael S. Branicky_, Mar 04 2021

%Y Cf. A019518, A046035.

%K nonn,base,hard,more

%O 1,1

%A _James G. Merickel_, Sep 25 2012

%E a(10) from _Charles R Greathouse IV_, Sep 27 2012

%E a(11)-a(12) from _Michael S. Branicky_, Jul 27 2023

%E a(13) from _Michael S. Branicky_, Aug 03 2023