A362429 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.

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

Offset

1,2

Comments

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

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.

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.

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

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

Links

Table of n, a(n) for n=1..9.

Example

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

Prog

(Python)
from gmpy2 import is_prime
from sympy.ntheory import digits
from itertools import count, islice
def c(base, s=""):
    if base == 1: yield from (s:=s+"1"*n for n in count(1))
    else:
        yield from (s:=s+"".join(map(str, digits(n, base)[1:])) for n in count(1))
def a(n):
    if n == 1: return -1
    return next(k for k, t in enumerate(c(n), 1) if is_prime(int(t)))

Crossrefs

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

Sequence in context: A029569 A152102 A183663 * A201957 A152500 A066370

Adjacent sequences: A362426 A362427 A362428 * A362430 A362431 A362432

Keyword

sign,base,more

Author

Michael S. Branicky and N. J. A. Sloane, Apr 19 2023

Status

approved