Description
Given an integer n, return the decimal value of the binary string formed by concatenating the binary representations of 1 to n in order, modulo 109 + 7.
Example 1:
Input: n = 1 Output: 1 Explanation: "1" in binary corresponds to the decimal value 1.
Example 2:
Input: n = 3 Output: 27 Explanation: In binary, 1, 2, and 3 corresponds to "1", "10", and "11". After concatenating them, we have "11011", which corresponds to the decimal value 27.
Example 3:
Input: n = 12 Output: 505379714 Explanation: The concatenation results in "1101110010111011110001001101010111100". The decimal value of that is 118505380540. After modulo 109 + 7, the result is 505379714.
Constraints:
1 <= n <= 105
Solutions
This function concatenates the binary representations of all numbers from 1 to n into a single binary number and returns the result modulo 1e9 + 7. Here's how it works: it loops through each number i from 1 to n, calculates how many bits are needed to represent i using Math.log2(i), then shifts the accumulated result left by that many bit positions (using result * (1 << bitLength)) and adds i to it. For example, if n=3, it concatenates binary 1 + binary 10 + binary 11 = "11011" (which equals 27 in decimal). The modulo operation (% MOD) keeps the number from overflowing by taking the remainder when divided by 1e9 + 7.
/**
* @param {number} n
* @return {number}
*/
var concatenatedBinary = function (n) {
const MOD = 1e9 + 7;
let result = 0;
for (let i = 1; i <= n; i++) {
const bitLength = Math.floor(Math.log2(i)) + 1;
result = ((result * (1 << bitLength)) % MOD + i) % MOD;
}
return result;
};