1219 — the integer and the year AD

Mathematical properties

In arithmetic, 1219 = 23 × 53, the product of two distinct prime numbers, so it is a semiprime. Its positive divisors are 1, 23, 53 and 1219, giving the divisor-count τ(1219) = 4. The sum of all divisors is σ(1219) = 1 + 23 + 53 + 1219 = 1296, which equals 6⁴; the sum of proper divisors is 77, so 1219 is a deficient number. Euler's totient satisfies φ(1219) = (23 − 1)(53 − 1) = 1144. The Möbius function μ(1219) = (+1) because 1219 is squarefree with two distinct prime factors.

Representations and simple digital properties: in base 10 the digit sum is 1 + 2 + 1 + 9 = 13; in binary 1219 is 1001100011₂, in hexadecimal it is 4C3₁₆, and in octal it is 2303₈. It is neither a perfect square nor a perfect cube, and it is not a triangular number in the usual sense.

Year AD 1219 — historical context and notable events

The year 1219 falls in the High Middle Ages. Broadly known developments often associated with this year include the opening of large-scale Mongol operations against the Khwarezmian Empire under Genghis Khan, which began the westward expansion of Mongol military campaigns across Central Asia, and events connected with the Fifth Crusade in the eastern Mediterranean. In the course of the crusading operations, Crusader forces undertook the campaign against the Egyptian port of Damietta; contemporaneous accounts also record Francis of Assisi travelling to the region and meeting Muslim leaders while the crusade was under way. These occurrences are commonly cited in general historical treatments of the period.

Uses, examples and distinctions

As with many integers, 1219 appears routinely as an identifier: catalogue numbers, part or model numbers, room numbers, and serial labels. Its elementary number‑theory attributes — semiprime, small divisor set, totient and Möbius values — make it a convenient classroom example when illustrating multiplicative arithmetic functions and properties of composite numbers.

Further notes

• 1219 is written MCCXIX in Roman numerals.
• It is a semiprime and therefore squarefree, so multiplicative functions factor simply.
• The coincidence σ(1219) = 6⁴ is a notable numerical curiosity but does not signal a standard classification beyond the computed identity.