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Consider decomposition the factorial n! into multiplicative factors p_k^(b_k) arranged in nondecreasing order. For example, 4! = 3·2^3 (1) = 2·3·4 (2) = 2·2·2·3 (3) and 5! = ...

Based on methods developer in collaboration with M. Leclert, Catalan (1865) computed the constant K=0.915965594177... (OEIS A006752) now known as Catalans' constant to 9 ...

The Earls sequence gives the starting position in the decimal digits of pi (or in general, any constant), not counting digits to the left of the decimal point, at which a ...

A method for finding roots of a polynomial equation f(x)=0. Now find an equation whose roots are the roots of this equation diminished by r, so (1) The expressions for f(r), ...

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by 668+866=1534, so the next ...

A Smith number is a composite number the sum of whose digits is the sum of the digits of its prime factors (excluding 1). (The primes are excluded since they trivially ...

Let sopfr(n) be the sum of prime factors (with repetition) of a number n. For example, 20=2^2·5, so sopfr(20)=2+2+5=9. Then sopfr(n) for n=1, 2, ... is given by 0, 2, 3, 4, ...

An encoding is a way of representing a number or expression in terms of another (usually simpler) one. However, multiple expressions can also be encoded as a single ...

In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given ...

The expectation value of a function f(x) in a variable x is denoted <f(x)> or E{f(x)}. For a single discrete variable, it is defined by <f(x)>=sum_(x)f(x)P(x), (1) where P(x) ...