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Legendre's formula counts the number of positive integers less than or equal to a number x which are not divisible by any of the first a primes, (1) where |_x_| is the floor ...

Let n be a positive integer and r(n) the number of (not necessarily distinct) prime factors of n (with r(1)=0). Let O(m) be the number of positive integers <=m with an odd ...

Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, ...

A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, ...

A number x is said to have "n figures" if it takes n digits to express it. The number of figures is therefore equal to one more than the power of 10 in the scientific ...

The smallest composite squarefree number (2·3), and the third triangular number (3(3+1)/2). It is the also smallest perfect number, since 6=1+2+3. The number 6 arises in ...

There exist a variety of formulas for either producing the nth prime as a function of n or taking on only prime values. However, all such formulas require either extremely ...

A polyhedral graph is completely regular if the dual graph is also regular. There are only five types. Let rho be the number of graph edges at each node, rho^* the number of ...

The global clustering coefficient C of a graph G is the ratio of the number of closed trails of length 3 to the number of paths of length two in G. Let A be the adjacency ...

Consider a game where everyone in a classroom picks a number between 0 and 100. The person closest to half the average wins. Obviously, picking a number over 50 would be ...