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An Achilles number is a positive integer that is powerful (in the sense that each prime factor occurs with exponent greater than one) but imperfect (in the sense that the ...

A positive integer which is divisible by the sum of its digits, also called a Niven number (Kennedy et al. 1980) or a multidigital number (Kaprekar 1955). The first few are ...

The number one (1), also called "unity," is the first positive integer. It is an odd number. Although the number 1 used to be considered a prime number, it requires special ...

Let s(n)=sigma(n)-n, where sigma(n) is the divisor function and s(n) is the restricted divisor function, and define the aliquot sequence of n by ...

Consider the process of taking a number, taking its digit sum, then adding the digits of numbers derived from it, etc., until the remaining number has only one digit. The ...

An n-persistent number is a positive integer k which contains the digits 0, 1, ..., 9 (i.e., is a pandigital number), and for which 2k, ..., nk also share this property. No ...

To define a recurring digital invariant of order k, compute the sum of the kth powers of the digits of a number n. If this number n^' is equal to the original number n, then ...

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 double Mersenne number is a number of the form M_(M_n)=2^(2^n-1)-1, where M_n is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, ...

A figurate number which is the sum of two consecutive pyramidal numbers, O_n=P_(n-1)+P_n=1/3n(2n^2+1). (1) The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, ...