8064

8064 is a natural number that lies between 8063 and 8065. In mathematics, it is an even composite number with a rich factorization that reveals several properties.

Its prime factorization is 8064 = 2^7 × 3^2 × 7. From this factorization, the number of positive divisors is (7+1)(2+1)(1+1) = 48. The Euler totient function yields φ(8064) = 8064 × (1 − 1/2) × (1 − 1/3) × (1 − 1/7) = 2304, and the sum of its positive divisors is σ(8064) = 26520, so the sum of proper divisors is 18456.

8064 has the property of being a Harshad (Niven) number in base 10, since the sum of its digits is 8 + 0 + 6 + 4 = 18 and 8064 is divisible by 18, giving 8064 ÷ 18 = 448. The number is divisible by 128, 9, and 7, consistent with its factorization.

In numerical representations, 8064 is written as 1111110000000 in binary, 0x1F80 in hexadecimal, and 17600 in octal. A compact algebraic observation is that 8064 = 2^7 × (2^6 − 1) = 128 × 63, highlighting its composition from a power of two and a Mersenne component.