A361312 Smallest prime p such that the decimal expansion of p remains prime through exactly n iterations of base-10 to base-2 conversion (A007088).

2, 3, 5, 3893257, 9632552297

Offset: 0

Author

Ya-Ping Lu, Mar 08 2023

Keywords

• nonn
• base
• more

Comments

Prime numbers that remain primes after 1, 2, 3, and 4 iterations are A065720, A123266, A256621 and A256622, respectively.

Examples

			a(0) = 2 because prime number 2 in base 2 is 10, and 10 in base 10 is not a prime.
a(1) = 3 because 3 = 11_2 and 11_10 is a prime. In the second iteration, however, 11_10 = 1011_2 and 1011_10 is not a prime.
a(2) = 5 because 5 = 101_2 and 101_10 = 1100101_2. Both 101 and 1100101 are primes in base 10. In the third iteration, 1100101_10 = 100001100100101000101_2 and 100001100100101000101_10 is not a prime.
		

Links

• Carlos Rivera, Puzzle 280. 3893257, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A007088, A036952, A065720, A123266, A256621, A256622.

Programs

Python

from sympy import isprime, nextprime
p = 1; mx = 5; I = [*range(mx)]; R = [*range(mx)]
while I:
    p = nextprime(p); ct = 0; q = p
    while isprime(int(bin(q)[2:])): ct += 1; q = int(bin(q)[2:])
    if ct in I: R[ct] = p; I.remove(ct)
print(*R, sep = ", ")