## Square subsets

Published on Saturday, 27th January 2018, 10:00 pm; Solved by 307;
Difficulty rating: 45%

### Problem 619

For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of non-empty subsets in which the product of all elements is a perfect square.

For example $C(5,10)=3$, since the products of all elements of $\{5, 8, 10\}$, $\{5, 8, 9, 10\}$ and $\{9\}$ are perfect squares, and no other subsets of $\{5, 6, 7, 8, 9, 10\}$ have this property.

You are given that $C(40,55) =15$, and $C(1000,1234) \text{ mod } 1000000007=975523611$.

Find $C(1000000,1234567) \text{ mod } 1000000007$.