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A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. Two numbers sharing the property form a friendly pair. Larger clubs of mutually friendly numbers also exist. A number without such friends is called solitary. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
A composite number is a positive integer which has a positive divisor other than one or itself. ...
A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ...
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. ...
An Achilles number is a number that is powerful but not a perfect power. ...
In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...
In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function σ(n)) is equal to 2n _ 1. ...
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. ...
In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. ...
In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i. ...
A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. ...
In mathematics, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. ...
In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a natural number that has no semiperfect proper divisor. ...
A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ...
In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ...
In mathematics, a highly abundant number is a certain kind of natural number. ...
In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. ...
In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. ...
A highly composite number is a positive integer which has more divisors than any positive integer below it. ...
In mathematics, a superior highly composite number is a certain kind of natural number. ...
In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ...
The term weird number also refers to a phenomenon in twos complement arithmetic. ...
Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ...
Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ...
In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ...
In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number. ...
A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. ...
A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents). ...
An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents). ...
An extravagant number (also known as a wasteful number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). ...
Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ...
In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ...
In mathematics, factorization (British English: factorisation) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
The characteristic in question is the rational number σ(n) / n, in which σ denotes the divisor function. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...
The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming the friendly pair (6, 28), and even the friendly triplet (6, 28, 496). There are several unsolved problems related to the friendly numbers.
In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function. Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ...
Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ...
The divisor function
If n is a positive natural number, σ(n) is the sum of its divisors. For example, 10 is divisible by 1, 2, 5, and 10, and so σ(10) = 1 + 2 + 5 + 10 = 18.
Kinship and friendliness Define the "kinship" κ(n) of a positive natural number n as the rational number σ(n)/n. For example, κ(10) = 18/10 = 9/5. Numbers whose kinship equals 2 are also known as perfect numbers. The name "kinship" and notation κ(n) are not standard usage, and are introduced here solely for the ease of presentation. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...
Numbers are mutually friendly if they share their kinship. For example, κ(6) = κ(28) = κ(496) = 2. The numbers 6, 28 and 496 are all perfect, and therefore mutually friendly. As another example, (103, 476) is a friendly pair, since κ(103) = κ(476) = 36/17.
Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into "clubs" of mutually friendly numbers. In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ...
A partition of U into 6 blocks: a Venn diagram representation. ...
Solitary numbers The numbers that belong to a singleton club, because no other number is friendly, are the solitary numbers. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, whenever the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – the number n is solitary. For a prime number p we have σ(n) = p + 1, which is co-prime with p. Coprime - Wikipedia /**/ @import /skins-1. ...
In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...
No general method is known for determining whether a number is friendly or solitary. The smallest number whose classification is unknown (as of 2007) is 10; it is conjectured to be solitary; if not, its smallest friend is a fairly large number.
Large clubs It is an open problem whether there are infinitely large clubs of mutually friendly numbers. The perfect numbers form a club, and it is conjectured that there are infinitely many perfect numbers (at least as many as there are Mersenne primes), but no proof is known. Currently (as of 2007) 44 perfect numbers are known, so at least one club of mutually friendly numbers contains 44 members, the largest of which, when written out in decimal notation, is more than 19 million digits long. In mathematics, a Mersenne number is a number that is one less than a power of two. ...
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