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A pair of numbers m and n such that sigma^*(m)=sigma^*(n)=m+n, where sigma^*(n) is the unitary divisor function. Hagis (1971) and García (1987) give 82 such pairs. The first ...

A divisor d of n for which GCD(d,n/d)=1, (1) where GCD(m,n) is the greatest common divisor. For example, the divisors of 12 are {1,2,3,4,6,12}, so the unitary divisors are ...

An untouchable number is a positive integer that is not the sum of the proper divisors of any number. The first few are 2, 5, 52, 88, 96, 120, 124, 146, ... (OEIS A005114). ...

The simple process of voting leads to surprisingly counterintuitive paradoxes. For example, if three people vote for three candidates, giving the rankings A, B, C; B, C, A; ...

Analysis

The quantity twelve (12) is sometimes known as a dozen. It is in turn one twelfth of a gross. Base-12 is known as duodecimal. The Schoolhouse Rock segment "Little Twelvetoes" ...

A number traditionally associated with bad luck. A so-called baker's dozen is equal to 13. Fear of the number 13 is called triskaidekaphobia. There are 13 Archimedean solids. ...

Some interesting properties (as well as a few arcane ones not reiterated here) of the number 239 are discussed in Schroeppel (1972). 239 appears in Machin's formula ...

Feynman (1997, p. 116) noticed the curious fact that the decimal expansion 1/(243)=0.004115226337448559... (1) repeats pairs of the digits 0, 1, 2, 3, ... separated by the ...

3 is the only integer which is the sum of the preceding positive integers (1+2=3) and the only number which is the sum of the factorials of the preceding positive integers ...