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Smallest k such that the concatenation of the numbers 123...k in base n is prime when interpreted as a decimal number, or -1 if no such prime exists.
2

%I #33 Sep 16 2024 12:38:54

%S -1,231,7315,3241,6,12891,22,227,127

%N Smallest k such that the concatenation of the numbers 123...k in base n is prime when interpreted as a decimal number, or -1 if no such prime exists.

%C The sequence can be extended to bases larger than 10 by concatenating the decimal equivalents of digits.

%C a(1) is -1 since no such primes are possible (the sequence in question is A362118). Proof. The number of ones in the resulting repunit is triangular and per A000217, 3 is the only prime triangular number, and per A004023, prime repunits must have prime indices.

%C If it exists, a(10) would be the index of the first prime in A007908. See A007908 for the latest information about the search for this prime.

%C a(10), ..., a(14) are respectively ?, 144, 307, ?, 25.

%C a(10) and a(13) are presently unknown. a(13) > 10000 if it is not -1.

%e a(5) is 6: 12341011 (concatenate 1 though 6 in base 5) is a prime when interpreted as a decimal number.

%o (Python)

%o from gmpy2 import is_prime

%o from sympy.ntheory import digits

%o from itertools import count, islice

%o def c(base, s=""):

%o if base == 1: yield from (s:=s+"1"*n for n in count(1))

%o else:

%o yield from (s:=s+"".join(map(str, digits(n, base)[1:])) for n in count(1))

%o def a(n):

%o if n == 1: return -1

%o return next(k for k, t in enumerate(c(n), 1) if is_prime(int(t)))

%Y Sequences of concatenations: A362118 (base 1), A058935 (base 2), A360502 (base 3), A117640 (base 4), A362117 (base 5), A362119 (base 6), A007908 (base 10).

%Y Cf. A376221.

%K sign,base,more

%O 1,2

%A _Michael S. Branicky_ and _N. J. A. Sloane_, Apr 19 2023