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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.
Forty is the fourth octagonal number. As the sum of the first four pentagonal numbers: , 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 with exactly nine solutions to the equation Euler's totient function (for values 41, 55, 75, 82, 88, 100, 110, 132, and 150 of ).
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 .
Swiss mathematician Leonard Euler noted forty prime numbers generated by the quadratic polynomial , with values :
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, and 1601.
The differences between terms are 0, 2, 4, 6, 8, ..., 78 (equivalently, up through a difference of twice 39). The first such prime (41) is the thirteenth prime number, where 13 divides the largest thrice over. The last such prime 1601 is the 252nd prime number (that is the sum between two through twenty-two, inclusive) as well as one more than the square of forty, 402 = 1600. Importantly, 41 is also the largest of six lucky numbers of Euler of the form,
These forty prime numbers are the same prime numbers that are generated using the polynomial with values of from 1 through 40, and are also known in this context as Euler's "lucky" numbers.
Given 40, the Mertens function returns 0, as with 39 — the only other smaller number to return a value of zero is 2. Adding 39 and 40 yields a prime number, the twenty-second indexed prime 79. It is equal to the sum of all six lucky numbers of Euler, specifically .
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