A powerful number is a positive integer m such that for every prime number p dividing m, p2 divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful; the following is a list of all powerful numbers between 1 and 1000: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.... If m = a2b3 every prime in the prime factorization of a appears in the prime factorization of m with an exponent of at least two, every prime in the prime factorization of b appears in the prime factorization of m with an exponent of at least three. In the other direction, suppose that m is powerful, with prime factorization m = ∏ p i α i, where each αi ≥ 2.
Define γi to be three if αi is odd, zero otherwise, define βi = αi − γi. All values βi are nonnegative integers, all values γi are either zero or three, so m = = 2 3 supplies the desired representation of m as a product of a square and a cube. Informally, given the prime factorization of m, take b to be the product of the prime factors of m that have an odd exponent; because m is powerful, each prime factor with an odd exponent has an exponent, at least 3, so m/b3 is an integer. In addition, each prime factor of m/b3 has an exponent, so m/b3 is a perfect square, so call this a2. For example: m = 21600 = 2 5 × 3 3 × 5 2, b = 2 × 3 = 6, a = m b 3 = 2 2 × 5 2 = 10, m = a 2 b 3 = 10 2 × 6 3; the representation m = a2b3 calculated in this way has the property that b is squarefree, is uniquely defined by this property. The sum of the reciprocals of the powerful numbers converges; the value of this sum may be written in several other ways, including as the infinite product ∏ p = ζ ζ ζ = 315 2 π 4 ζ, where p runs over all primes, ζ denotes the Riemann zeta function, ζ is Apéry's constant.
More the sum of the reciprocals of the sth powers of the powerful numbers is equal to ζ ζ ζ whenever it converges. Let k denote the number of powerful numbers in the interval. K is proportional to the square root of x. More c x 1 / 2 − 3 x 1 / 3 ≤ k ≤ c x 1 / 2, c = ζ / ζ = 2.173 …. The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation x2 − 8y2 = 1 has infinitely many
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, 11 = 6 + 3 + 2; the sequence of practical numbers begins 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150.... Practical numbers were used by Fibonacci in his Liber Abaci in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators; the name "practical number" is due to Srinivasan. He noted that "the subdivisions of money and measures involve numbers like 4, 12, 16, 20 and 28 which are supposed to be so inconvenient as to deserve replacement by powers of 10."
He rediscovered the number theoretical property of such numbers and was the first to attempt a classification of these numbers, completed by Stewart and Sierpiński. This characterization makes it possible to determine whether a number is practical by examining its prime factorization; every perfect number and every power of two is a practical number. Practical numbers have been shown to be analogous with prime numbers in many of their properties; the original characterisation by Srinivasan stated that a practical number cannot be a deficient number, one of which the sum of all divisors is less than twice the number unless the deficiency is one. If the ordered set of all divisors of the practical number n is d 1, d 2... D j with d 1 = 1 and d j = n Srinivasan's statement can be expressed by the inequality 2 n ≤ 1 + ∑ i = 1 j d i. In other words, the ordered sequence of all divisors d 1 < d 2 <... < d j of a practical number has to be a complete sub-sequence. This partial characterization was extended and completed by Stewart and Sierpiński who showed that it is straightforward to determine whether a number is practical from its prime factorization.
A positive integer greater than one with prime factorization n = p 1 α 1... P k α k is practical if and only if each of its prime factors p i is small enough for p i − 1 to have a representation as a sum of smaller divisors. For this to be true, the first prime p 1 must equal 2 and, for every i from 2 to k, each successive prime p i must obey the inequality p i ≤ 1 + σ = 1 + ∏ j = 1 i − 1 p j α j + 1 − 1 p j − 1, where σ denotes the sum of the divisors of x. For example, 2 × 32 × 29 × 823 = 429606 is practical, because the inequality above holds for each of its prime factors: 3 ≤ σ + 1 = 4, 29 ≤ σ + 1 = 40, 823 ≤ σ + 1 = 1171; the condition stated above is sufficient for a number to be practical. In one direction, this condition is necessary in order to be able to represent p i − 1 as a sum of divisors of n, because if the inequality failed to be true even adding together all the smaller divisors would give a sum too small to reach p i − 1. In the other direction, the condition is sufficient.
In mathematics, a perfect power is a positive integer that can be resolved into equal factors, whose root can be extracted. I.e. a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3 n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are considered perfect powers. A sequence of perfect powers can be generated by iterating through the possible values for k; the first few ascending perfect powers in numerical order are: 2 2 = 4, 2 3 = 8, 3 2 = 9, 2 4 = 16, 4 2 = 16, 5 2 = 25, 3 3 = 27, 2 5 = 32, 6 2 = 36, 7 2 = 49, 2 6 = 64, 4 3 = 64, 8 2 = 64, … The sum of the reciprocals of the perfect powers is 1: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = 1. Which can be proved as follows: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ∑ k = 0 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 = ∑ m = 2 ∞ 1 m = ∑ m = 2 ∞ = 1.
The first perfect powers without duplicates are:, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024... The sum of the reciprocals of the perfect powers p without duplicates is: ∑ p 1 p = ∑ k = 2 ∞ μ ≈ 0.874464368 … where μ is the Möbius function and ζ is the Riemann zeta function. According to Euler, Goldbach showed that the sum of 1/p − 1 over the set of perfect powers p, excluding 1 and excluding duplicates, is 1: ∑ p 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1; this is sometimes known as the Goldbach–Euler theorem. Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to k ≤ log 2 n. So if the divisors of n are n 1, n 2, …, n j one of the values n 1 2, n 2 2
Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic; the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this defines an infinite number of instances, it is done in such a way that no loop or infinite chain of references can occur. In mathematics and computer science, a class of objects or methods exhibit recursive behavior when they can be defined by two properties: A simple base case —a terminating scenario that does not use recursion to produce an answer A set of rules that reduce all other cases toward the base caseFor example, the following is a recursive definition of a person's ancestors: One's parents are one's ancestors; the ancestors of one's ancestors are one's ancestors. The Fibonacci sequence is a classic example of recursion: Fib = 0 as base case 1, Fib = 1 as base case 2, For all integers n > 1, Fib:= Fib + Fib.
Many mathematical axioms are based upon recursive rules. For example, the formal definition of the natural numbers by the Peano axioms can be described as: 0 is a natural number, each natural number has a successor, a natural number. By this base case and recursive rule, one can generate the set of all natural numbers. Recursively defined mathematical objects include functions and fractals. There are various more tongue-in-cheek "definitions" of recursion. Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be'recursive'. To understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules; the running of a procedure involves following the rules and performing the steps. An analogy: a procedure is like a written recipe. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure.
For instance, a recipe might refer to cooking vegetables, another procedure that in turn requires heating water, so forth. However, a recursive procedure is where one of its steps calls for a new instance of the same procedure, like a sourdough recipe calling for some dough left over from the last time the same recipe was made; this creates the possibility of an endless loop. If properly defined, a recursive procedure is not easy for humans to perform, as it requires distinguishing the new from the old invocation of the procedure. For this reason recursive definitions are rare in everyday situations. An example could be the following procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point. If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively. Whether this defines a terminating procedure depends on the nature of the maze: it must not allow loops. In any case, executing the procedure requires recording all explored branching points, which of their branches have been exhaustively tried.
Linguist Noam Chomsky among many others has argued that the lack of an upper bound on the number of grammatical sentences in a language, the lack of an upper bound on grammatical sentence length, can be explained as the consequence of recursion in natural language. This can be understood in terms of a recursive definition of a syntactic category, such as a sentence. A sentence can have a structure in which what follows the verb is another sentence: Dorothy thinks witches are dangerous, in which the sentence witches are dangerous occurs in the larger one. So a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, optionally another sentence; this is just a special case of the mathematical definition of recursion. This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, therefore many ways in which a sentence can embed instances of one
In mathematics, the Perrin numbers are defined by the recurrence relation P = P + P for n > 2,with initial values P = 3, P = 0, P = 2. The sequence of Perrin numbers starts with 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39... The number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number for n > 1. This sequence was mentioned implicitly by Édouard Lucas. In 1899, the same sequence was mentioned explicitly by François Olivier Raoul Perrin; the most extensive treatment of this sequence was given by Shanks. The generating function of the Perrin sequence is G = 3 − x 2 1 − x 2 − x 3. N = The Perrin sequence numbers can be written in terms of powers of the roots of the equation x 3 − x − 1 = 0; this equation has 3 roots. Given these three roots, the Perrin sequence analogue of the Lucas sequence Binet formula is P = p n + q n + r n. Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n.
For large n the formula reduces to P ≈ p n This formula can be used to calculate values of the Perrin sequence for large n. The ratio of successive terms in the Perrin sequence approaches p, a.k.a. the plastic number, which has a value of 1.324718. This constant bears the same relationship to the Perrin sequence as the golden ratio does to the Lucas sequence. Similar connections exist between p and the Padovan sequence, between the golden ratio and Fibonacci numbers, between the silver ratio and Pell numbers. From the Binet formula, we can obtain a formula for G in terms of G, G and G. Example magma code: P<x>:= PolynomialRing.
In number theory, the Padovan sequence is the sequence of integers P defined by the initial values P = P = P = 1, the recurrence relation P = P + P. The first few values of P are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265... The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan: Modern Primitive; the sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. He writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics"; the above definition is the one given by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. In the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence satisfies the recurrence relation P = P + P Starting from this, the defining recurrence and other recurrences as they are discovered, one can create an infinite number of further recurrences by replacing P by P + P The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.
This is a property of recurrence relations. The Perrin sequence can be obtained from the Padovan sequence by the following formula: P e r r i n = P + P; as with any sequence defined by a recurrence relation, Padovan numbers P for m<0 can be defined by rewriting the recurrence relation as P = P − P, Starting with m = −1 and working backwards, we extend P to negative indices: The sum of the first n terms in the Padovan sequence is 2 less than P i.e. ∑ m = 0 n P = P − 2. Sums of alternate terms, sums of every third term and sums of every fifth term are related to other terms in the sequence: ∑ m = 0 n P = P − 1 OEIS: A077855 ∑ m = 0 n P = P − 1 ∑ m = 0 n P = P OEIS: A034943 ∑ m = 0 n P = P − 1 ∑ m = 0 n P = P − 1 ∑ m = 0 n P = P. OEIS: A012772Sums involving products of terms in the Padovan sequence satisfy the following identities: ∑ m = 0 n P 2 = P 2 − P 2 − P 2 ∑ m = 0 n P 2 P ( m