On an indivisibility version of Iizuka's conjecture

Abstract: We establish the existence of a positive proportion of ( d \in \mathbb{N} ) such that the class numbers of ( \mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \dots, \mathbb{Q}(\sqrt{d+n}) ) are not divisible by ( 3^k ), where ( n = 3^{k+1} - 5 ) and ( k \in \mathbb{N} ). This provides an indivisibility analog of Iizuka's conjecture. Similarly, for ( d < 0 ) and the same ( n ), a positive proportion of ( d ) ensures that the class numbers of ( \mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}), \dots, \mathbb{Q}(\sqrt{d+n}) ) are not divisible by ( 3^{k+1} ). Denoting the square-free natural numbers by ( (d_n) ), we prove the existence of a positive density set of ( i\in\mathbb{N} ) such that the class numbers of ( \mathbb{Q}(\sqrt{d_i}), \mathbb{Q}(\sqrt{d_{i+1}}), \dots, \mathbb{Q}(\sqrt{d_{i+n}}) ) are not divisible by ( 3^k ). Moreover, we study the indivisibility of class numbers by $3$ for imaginary biquadratic fields.