4 problems

Let the integer sequence be the variant of the happy-numbers sequence defined in the paper, and let $k$ be one of its terms. Three-step convergence conjecture. The integer $k$ conv…

Let $\gamma_2(\eta)$ denote the sequence introduced in the paper, and let a Mersenne prime be a prime number of the form $2^p-1$, where $p$ is prime. Mersenne-primality conjecture.…

Conjecture. For every even integer $b\leq -12$, there exist arbitrarily long finite sequences of consecutive $b$-happy numbers.

Consecutive-sequence conjecture. For each $u\in U_{[c,b]}$, there exist arbitrarily long finite sequences of consecutive $u$-attracted numbers for $S_{[c,b]}$.