Overview
1416 is a natural number that follows 1415 and precedes 1417. It is an even composite integer whose arithmetic and combinatorial properties are straightforward to derive from its prime factorization. As with any whole number, 1416 can be considered in contexts ranging from elementary number theory to date notation for historical years.
Mathematical characteristics
Prime factorization: 1416 = 2^3 · 3 · 59. From this factorization one can deduce several standard arithmetic invariants.
Divisor-related data: The number has (3+1)(1+1)(1+1) = 16 positive divisors. The sum of all positive divisors (the sigma function) equals 3600, so the sum of proper divisors is 3600 − 1416 = 2184. Because the sum of proper divisors exceeds the number itself, 1416 is classified as an abundant number (abundance 2184 − 1416 = 768). Euler's totient function gives φ(1416) = 464.
Divisors
- All positive divisors: 1, 2, 3, 4, 6, 8, 12, 24, 59, 118, 177, 236, 354, 472, 708, 1416.
Representations and classifications
1416 is not a prime, not a perfect square, and not a triangular or pentagonal figurate number. In common numeral systems it is represented as:
- binary: 10110001000
- octal: 2610
- hexadecimal: 0x588
- Roman numerals: MCDXVI
Historical and contextual notes
As a year number, 1416 may refer to 1416 AD (MCDXVI) in the Julian calendar or to 1416 BC when using the long historical timescale; each usage places the numeral in very different historical contexts. Mathematically, 1416 serves as an ordinary illustrative example when explaining factorization, divisor functions, abundant numbers, and base conversions in elementary number theory and arithmetic courses.
Because its prime factors include a relatively large prime (59) alongside small primes (2 and 3), 1416 is useful in exercises that show how a single larger factor affects divisor counts, totient values, and the structure of multiplicative functions.