Concept

# 40 (number)

Summary
40 (forty) is the natural number following 39 and preceding 41. Though the word is related to four (4), the spelling forty replaced fourty during the 17th century and is now the standard form. In mathematics Forty is the fourth octagonal number. As the sum of the first four pentagonal numbers: 1 + 5 + 12 + 22 =40, it is also is the fourth pentagonal pyramidal number. Forty is a repdigit in ternary, and a Harshad number in decimal. 40 is the smallest number n with exactly nine solutions to the equation Euler's totient function \varphi (x)=n (for values 41, 55, 75, 82, 88, 100, 110, 132, and 150 of x). 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. 40 is also the ninth refactorable number. Forty is the number of n-queens problem solutions for n=7. Euler's lucky numbers Swiss mathematician Leonard Euler n
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