This is a semigroup: if m and n are in the sequence, then so is m*n.
If n is in the sequence and is divisible by prime p, then so is p*n.
The only prime powers in the sequence are the powers of 5.
Conjecture: Every member of the sequence except 1 is of the form p*m where p is prime and m is in the sequence. (End)
There are infinitely many primes p that divide some term in the sequence. Proof: Define the set A as all primes p such that a k where p divides 2^(5^k) + 3^(5^k) exists is finite. Since 2^(5^(k+1)) + 3^(5^(k+1)) is 2^(5^k) + 3^(5^k) multiplied by some positive integer a. It can be verified that gcd(a, 2^(5^k) + 3^(5^k)) is 5, so 2^(5^(k+1)) + 3^(5^(k+1)) has a larger number of different prime factors than 2^(5^k) + 3^(5^k). Therefore, A is infinite. For each prime q in A, suppose that q divides 2^(5^x) + 3^(5^x) for some x, then 5^x also divides it, so 5^x*q divides it as well, hence 5^x*q is a term of the sequence. The original theorem is proved. - Yifan Xie, Nov 14 2024
