The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few numbers in the sequence are

0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, ...

**Combinatorial interpretation**

These numbers can be used to solve several problems in combinatorial enumeration. The nth number in the sequence (starting with the number 0 for n = 0) counts

* The number of unordered rooted trees with n leaves in which all nodes including the root have either zero or exactly two children. These trees have been called Otter trees, after the work of Richard Otter on their combinatorial enumeration. They can also be interpreted as unlabeled and unranked dendrograms with the given number of leaves.

* The number of unordered rooted trees with n nodes in which the root has degree zero or one and all other nodes have at most two children. Trees in which the root has at most one child are called planted trees, and the additional condition that the other nodes have at most two children defines the weakly binary trees. In chemical graph theory, these trees can be interpreted as isomers of polyenes with a designated leaf atom chosen as the root.

* The number of different ways of organizing a single-elimination tournament for n players (with the player names left blank, prior to seeding players into the tournament). The pairings of such a tournament may be described by an Otter tree.

* The number of different results that could be generated by different ways of grouping the expression

**Formula**

The Wedderburn–Etherington numbers may be calculated using the recurrence relation

beginning with the base case

In terms of the interpretation of these numbers as counting rooted binary trees with n leaves, the summation in the recurrence counts the different ways of partitioning these leaves into two subsets, and of forming a subtree having each subset as its leaves. The formula for even values of n is slightly more complicated than the formula for odd values in order to avoid double counting trees with the same number of leaves in both subtrees.

]]>In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity. This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.

Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like.

or equivalently,

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

The silver rectangle is connected to the regular octagon.

]]>The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.23606797749978969640917366873127623544061835961152572427089...

which can be rounded down to 2.236 to within 99.99% accuracy. The approximation

161/72 (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than

1/10,000

approx.

The successive partial evaluations of the continued fraction, which are called its convergents, approach

:Their numerators are 2, 9, 38, 161, … , and their denominators are 1, 4, 17, 72, … .

Each of these is a best rational approximation of

; in other words, it is closer to than any rational with a smaller denominator.]]>The decimal value of the natural logarithm of 2 is approximately

The logarithm of 2 in other bases is obtained with the formula

The common logarithm in particular is

The inverse of this number is the binary logarithm of 10:

.By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.

]]>In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers: 12 and

The second of these has 76 decimal digits:6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.

]]>The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as

or . It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.As of December 2013, its numerical value in decimal notation had been computed to at least ten billion digits. Its decimal expansion, written here to 65 decimal places, is :

1.732050807568877293527446341505872366942805253810380628055806

The fraction

(1.732142857...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than

1/10,000

(approximately with a relative error of ). The rounded value of 1.732 is correct to within 0.01% of the actual value.

The fraction

(1.73205080756...) is accurate to

Archimedes reported a range for its value:

.The lower limit 1351/780 is an accurate approximation for

to 1/608400(six decimal places, relative error ) and the upper limit 265/153 to 2/23409

(four decimal places, relative error).]]>

The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as

, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. Its numerical value truncated to 65 decimal places is:

1.41421356237309504880168872420969807856967187537694807317667973799...

**The square root of 2.**

Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of electronic calculators and computers. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000

]]>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.

Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers".

A number that is not part of any friendly pair is called solitary.

The "abundancy" index of n is the rational number

, in which denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that . "Abundancy" is not the same as abundance, which is defined as ."Abundancy" may also be expressed as

where denotes a divisor function with equal to the sum of the k-th powers of the divisors of n.The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy"

the same as . The shared value 2 is an integer in this case but not in many other cases. Numbers with "abundancy" 2 are also known as perfect numbers. 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.

]]>In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using p-adic numbers. The p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.

More formally, for a given prime p, the field

of p-adic numbers is a completion of the rational numbers. The field is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility.The p in "p-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "p-adic" comes from the ending found in words such as dyadic or triadic.

]]>In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.

The first decimal Smarandache–Wellin numbers are:

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ...

**Smarandache–Wellin prime**

A Smarandache–Wellin number that is also prime is called a Smarandache–Wellin prime. The first three are 2, 23 and 2357 (sequence A069151 in the OEIS). The fourth is 355 digits long: it is the result of concatenating the first 128 prime numbers, through 719.

The primes at the end of the concatenation in the Smarandache–Wellin primes are

2, 3, 7, 719, 1033, 2297, 3037, 11927, ...

The indices of the Smarandache–Wellin primes in the sequence of Smarandache–Wellin numbers are:

1, 2, 4, 128, 174, 342, 435, 1429, ...

The 1429th Smarandache–Wellin number is a probable prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein in 1998. If it is proven prime, it will be the eighth Smarandache–Wellin prime. In March 2009, Weisstein's search showed the index of the next Smarandache–Wellin prime (if one exists) is at least 22077.

]]>In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number.

The first few perfect totient numbers are

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, .....

In symbols, one writes

for the iterated totient function. Then if c is the integer such that

one has that n is a perfect totient number if

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as

or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1.

Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring

).]]>In number theory, a Wieferich prime is a prime number p such that

divides therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.

As of March 2021, the only known Wieferich primes are 1093 and 3511

]]>In mathematics, a Riesel number is an odd natural number k for which

is composite for all natural numbers n (sequence A101036 in the OEIS). In other words, when k is a Riesel number, all members of the following set are composite:If the form is instead

, then k is a Sierpinski number.**Known Riesel numbers**

The sequence of currently known Riesel numbers begins with:

509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ...

]]>In number theory, a Sierpiński number is an odd natural number k such that

is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.In other words, when k is a Sierpiński number, all members of the following set are composite:

If the form is instead

, then k is a Riesel number.**Known Sierpiński numbers**

The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ...

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form

have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. Most currently known Sierpiński numbers possess similar covering sets.]]>A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes.

In base 10, all the permutable primes with fewer than 49,081 digits are known

Of the above, there are 16 unique permutation sets, with smallest elements

Note

is a repunit, a number consisting only of n ones (in base 10). Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proven that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.

There is no n-digit permutable prime for

which is not a repunit. It is conjectured that there are no non-repunit permutable primes other than those listed above.]]>