Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s(a)=b and s(b)=a, where s(n)=σ(n)-n is equal to the sum of positive divisors of n except n itself (see also divisor function).
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
21 (twenty-one) is the natural number following 20 and preceding 22.
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.
In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors of the number is greater than the number, but no subset of those divisors sums to the number itself.
70 (seventy) is the natural number following 69 and preceding 71.
90 (ninety) is the natural number following 89 and preceding 91.
34 (thirty-four) is the natural number following 33 and preceding 35.
58 (fifty-eight) is the natural number following 57 and preceding 59.
In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi, who observed that both 2 and 5 are untouchable.
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same abundancy form a friendly n-tuple.
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers.
177 is the natural number following 176 and preceding 178.
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.
In number theory, the aliquot sums(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is,
Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (m, n) are a pair of betrothed numbers if s(m) = n + 1 and s(n) = m + 1, where s(n) is the aliquot sum of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function.
Paul Poulet (1887–1946) was a self-taught Belgian mathematician who made several important contributions to number theory, including the discovery of sociable numbers in 1918. He is also remembered for calculating the pseudoprimes to base two, first up to 50 million in 1926, then up to 100 million in 1938. These are now often called Poulet numbers in his honour. In 1925, he published forty-three new multiperfect numbers, including the first two known octo-perfect numbers. His achievements are particularly remarkable given that he worked without the aid of modern computers and calculators.
