Swiss mathematician Leonard Euler noted forty prime numbers generated by the quadratic polynomial n 2 + n + 41 {\displaystyle n Though the word is related to four (4), the spelling forty replaced fourty during the 17th century [1] [2] and is now the standard form.

Adding up some subsets of the divisors of 40 (e.g., 1, 4, 5, 10, and 20) gives 40; hence, 40 is the ninth semiperfect number. [6] 40 is also the ninth refactorable number. [7] Forty is the fourth octagonal number. [3] As the sum of the first four pentagonal numbers: 1 + 5 + 12 + 22 = 40 {\displaystyle 1+5+12+22=40} , it is also the fourth pentagonal pyramidal number. [4] Forty is a repdigit in ternary, and a Harshad number in decimal. [5] is the smallest number n {\displaystyle n} with exactly nine solutions to the equation Euler's totient function φ ( x ) = n {\displaystyle \varphi (x)=n} (for values 41, 55, 75, 82, 88, 100, 110, 132, and 150 of x {\displaystyle x} ).