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42
Fortytwo (42) is a pronic number[1] and an abundant number; its prime factorization ( 2 Γ 3 Γ 7 {\displaystyle 2\times 3\times 7}) makes it the second sphenic number and also the second of the form ( 2 Γ 3 Γ r {\displaystyle 2\times 3\times r}).
Additional properties of the number 42 include:
It is the number of isomorphism classes of all simple and oriented directed graphs on 4 vertices. In other words, it is the number of all possible outcomes (up to isomorphism) of a tournament consisting of 4 teams where the game between any pair of teams results in three possible outcomes: the first team wins, the second team wins, or there is a draw. The group stage of the FIFA World cup is a good example.
It is the third primary pseudoperfect number.[2]
It is a Catalan number.[3] Consequently, 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered binary trees with six leaves, the number of ways in which five pairs of nested parentheses can be arranged, etc.
It is an alternating sign matrix number, that is, the number of 4by4 alternating sign matrices.
It is the smallest number k that is equal to the sum of the nonprime proper divisors of k, i.e., 42 = 1 + 6 + 14 + 21.
It is the number of partitions of 10βthe number of ways of expressing 10 as a sum of positive integers (note a different sense of partition from that above).
1111123, one of the 42 unordered integer partitions of 10 has 42 ordered compositions, since 7!/5!=42.
The angle of 42 degrees can be constructed with only compass and straight edge and using the golden ratio in 18 degree, i.e. the difference between constructible angles 60 and 18.
The 3 Γ 3 Γ 3 simple magic cube with rows summing to 42
Given 27 samesize cubes whose nominal values progress from 1 to 27, a 3 Γ 3 Γ 3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of three numbers whose sum of values is 42.
It is the third pentadecagonal number.[4] It is a meandric number and an open meandric number.
42 is the only known value that is the number of sets of four distinct positive integers a, b, c, d, each less than the value itself, such that ab β cd, ac β bd, and ad β bc are each multiples of the value. Whether there are other values remains an open question.[5]
42 is a (2,6)perfect number (supermultiperfect), as Ο2(n) = Ο(Ο(n)) = 6n.[6]
42 is the resulting number of the original Smith number ( 4937775 = 3 Γ 5 Γ 5 Γ 65837 {\displaystyle 4937775=3\times 5\times 5\times 65837}): Both the sum of its digits ( 4 + 9 + 3 + 7 + 7 + 7 + 5 {\displaystyle 4+9+3+7+7+7+5}) and the sum of the digits in its prime factorization ( 3 + 5 + 5 + ( 6 + 5 + 8 + 3 + 7 ) {\displaystyle 3+5+5+(6+5+8+3+7)}) result in 42.
The dimension of the Borel subalgebra in the exceptional Lie algebra e6 is 42.
42 is the largest number n such that there exist positive integers p, q, r with 1 = 1/n + 1/p + 1/q + 1/r
42 is the smallest number k such that for every Riemann surface C of genus g β₯ 2 {\displaystyle g\geq 2}, #Aut(C) β€ k deg(KC) = k(2g β 2) (Hurwitz's automorphisms theorem)
42 is the sum of the first six positive even numbers.
42 was the last natural number less than 100 whose representation as a sum of three cubes was found (in 2019). The representation is: ( β 80538738812075974 ) 3 + 80435758145817515 3 + 12602123297335631 3 {\displaystyle (80538738812075974)^{3}+80435758145817515^{3}+12602123297335631^{3}}.[7]
42 is a Harshad number in base 10, because the sum of the digits 4 and 2 is 6 (4 + 2 = 6), and 42 is divisible by 6.
42 is the number of ways to arrange the numbers 1 to 9 in a 3x3 matrix such that the numbers in each row and column are in ascending order.
42 is also ten factorial divided by the number of seconds in a day (i.e. 86400).
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