A348226 a(n) is the smallest positive integer that when expressed in bases 2 to n, but read in base n, is always prime.

2, 2, 43, 2, 45481, 2, 65484343, 186914543201, 50006393431, 2

Offset: 2

Author

Jesús Bellver Arnau, Oct 09 2021

Keywords

• nonn
• base
• more

Comments

a(n)=2 whenever n is prime.

Proof:

Let n be a prime number.

2 expressed in any base larger than 2 is still 2, which is prime.

2 expressed in base 2 is 10. And 10 read in base n is 1*n + 0 = n, which is prime.

The sequence, even when prime indexes are omitted, is not necessarily increasing.

Proof: a(9) > a(10).

Examples

			a(4) = 43, because
  43 is prime
  43 in base 3 is 1121 = 1*3^3 + 1*3^2 + 2*3 + 1 and
                         1*4^3 + 1*4^2 + 2*4 + 1 = 89, which is prime;
  43 in base 2 is 101011 = 1*2^5 + 0*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 1 and
                           1*4^5 + 0*4^4 + 1*4^3 + 0*4^2 + 1*4^1 + 1 = 1093, which is prime;
and 43 is the smallest positive integer with this property.
a(10) = 50006393431
      = 153060758677_9
      = 564447201127_8
      = 3420130221331_7
      = 34550030320411_6
      = 1304403114042211_5
      = 232210213100021113_4
      = 11210002000211222202121_3
      = 101110100100100111010000001001010111_2;
if we read these numbers as base-10 numbers, they are all prime. And 50006393431 is the smallest positive integer with this property.
		

Programs

PARI

isok(k, n) = {for (b=2, n, if (! ispseudoprime(fromdigits(digits(k, b), n)), return (0));); return (1);}
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 09 2021

Python

from gmpy2 import digits, is_prime, next_prime
def A348226(n): # code assumes n <= 63 or n is prime
    if is_prime(n):
        return 2
    p = 2
    while True:
        for i in range(n-1,1,-1):
            s = digits(p,i)
            if not is_prime(int(s,n)):
                break
        else:
            return p
        p = next_prime(p) # Chai Wah Wu, Nov 19 2021