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Lehmer's totient problem asks if there exist any composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function? No such numbers are known. However, any ...

A number n is called k-hyperperfect if n = 1+ksum_(i)d_i (1) = 1+k[sigma(n)-n-1], (2) where sigma(n) is the divisor function and the summation is over the proper divisors ...

Smarandache sequences are any of a number of simply generated integer sequences resembling those considered in published works by Smarandache such as the consecutive number ...

An NSW number (named after Newman, Shanks, and Williams) is an integer m that solves the Diophantine equation 2n^2=m^2+1. (1) In other words, the NSW numbers m index the ...

Let omega(n) be the number of distinct prime factors of n. If Psi(x) tends steadily to infinity with x, then lnlnx-Psi(x)sqrt(lnlnx)<omega(n)<lnlnx+Psi(x)sqrt(lnlnx) for ...

If alpha is any number and m and n are integers, then there is a rational number m/n for which |alpha-m/n|<=1/n. (1) If alpha is irrational and k is any whole number, there ...

A process of successively crossing out members of a list according to a set of rules such that only some remain. The best known sieve is the sieve of Eratosthenes for ...

A generalization of the Fibonacci numbers defined by 1=G_1=G_2=...=G_(c-1) and the recurrence relation G_n=G_(n-1)+G_(n-c). (1) These are the sums of elements on successive ...

Let s_b(n) be the sum of the base-b digits of n, and epsilon(n)=(-1)^(s_2(n)) the Thue-Morse sequence, then product_(n=0)^infty((2n+1)/(2n+2))^(epsilon(n))=1/2sqrt(2).

A binomial number is a number of the form a^n+/-b^n, where a,b, and n are integers. Binomial numbers can be factored algebraically as ...