153 (number)

153 is a natural number between 152 and 154. It is an odd number and not prime; it factors as 3 × 3 × 17, meaning its divisors are 1, 3, 9, 17, 51, and 153.

The number is notable for several mathematical properties that occur together simultaneously, particularly in relation to triangular numbers and digit-based sequences. It also appears in religious texts and historical interpretations.

Mathematical properties

153 is a triangular number: it is the sum of the integers from 1 to 17, that is, 1 + 2 + 3 + ... + 17 = 153. It is also a hexagonal number, since the formula n(2n − 1) when n = 9 gives 9 × 17 = 153.

153 is an Armstrong number (also called a narcissistic number) in base 10: if each digit is cubed and the results are added together, the sum equals the number itself — 1³ + 5³ + 3³ = 1 + 125 + 27 = 153.

Another property is that 153 equals the sum of the factorials of the first five numbers: 1! + 2! + 3! + 4! + 5! = 1 + 2 + 6 + 24 + 120 = 153.

There is a related convergence property: if one takes any number divisible by 3, cubes each digit and adds them together, then repeats this operation repeatedly, the result eventually reaches 153 and remains there.

Other uses

153 appears in Christian religious texts, notably in the Gospel account of the miraculous catch of fish in John chapter 21, where the number of fish caught is given as 153. Interpretations of this number vary, and there is no agreed-upon explanation.