OEIS

If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle 4, 16, 37, 58, 89, 145, 42, 20, 4, ... To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most 81m. For < ...

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Happy number, Happy number - Sequence behavior, Happy number - Happy Numbers in Other Bases

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The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivative, and therefore, its own primitive: and , where C is the arbitrary constant of integration. It is known that e is irrational (proof) and even more, transcendental (proof). It was the first number to be proved transcendental without having been specifically constructed for this purpose (cf. Liouville number); the proof w ...

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E mathematical constant, E mathematical constant - Definitions, E mathematical constant - Properties, E mathematical constant - History, E mathematical constant - Non-mathematical uses of e, E mathematical constant - Notes

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Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also Harshad numbers. But for the purpose of determining the Harshadness of n, the digits of n can only be added up once and n must be divisible by that sum; otherwise, it is not a Harshad number. For example, 99, although divisible by 9 as shown by 9 + 9 = 18 and 1 + 8 = 9, is not a Harshad number, since 9 + 9 = 18 = 2 × 32, and 99 is not divisible by 2. The base number will always be a Harshad numb ...

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Harshad number, Harshad number - What numbers can be Harshad numbers?, Harshad number - Consecutive Harshad numbers, Harshad number - Estimating the density of Harshad numbers

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These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F(n) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted tw ...

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Fibonacci number, Fibonacci number - Origins, Fibonacci number - The bee ancestry code, Fibonacci number - Relation to the golden ratio, Fibonacci number - Matrix form, Fibonacci number - Computation, Fibonacci number - Applications, Fibonacci number - Fibonacci numbers in nature, Fibonacci number - Identities, Fibonacci number - Common factors, Fibonacci number - Power series, Fibonacci number - Reciprocal sum constant, Fibonacci number - Generalizations, Fibonacci number - Vector space, Fibonacci number - Similar integer sequences, Fibonacci number - Other generalizations, Fibonacci number - Fibonacci primes, Fibonacci number - Fibonacci strings, Fibonacci number - Fiction, Fibonacci number - Journals

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Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n > 0 in base b ≥ 2, where it is written in standard notation with k+1 digits ai as: with, as usual, 0 ≤ ai < b for all i and ak ≠ 0. Then n is pa ...

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Palindromic number, Palindromic number - Formal definition, Palindromic number - Decimal palindromic numbers, Palindromic number - Other bases

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If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal. Allowing leading zeros, the sequence of cyclic numbers begins: 142857 0588235294117647 052631578947368421 0434782608695652173913 0344827586206896551724137931 0212765957446808510638297872340425531914893617 0169491525423728813559322033898305084745762711864406779661 01639344262295081967 ...

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Cyclic number, Cyclic number - Special cases, Cyclic number - Relation to recurring decimals, Cyclic number - Form of cyclic numbers, Cyclic number - Construction of cyclic numbers, Cyclic number - Other numeric bases, Cyclic number - External link

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The first conjecture made by Conway and Norton was the so-called "moonshine conjecture"; it states that there is an infinite-dimensional graded M-module with for all m, where From this it follows that every element g of M acts on each Vm and has character value which can be used to construct the McKay-Thompson series of g: The second conjecture of Conway and Norton then states t ...

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Monstrous moonshine, Monstrous moonshine - Formal versions of Conway's and Norton's conjectures, Monstrous moonshine - The Monster module, Monstrous moonshine - Borcherds' proof, Monstrous moonshine - Why monstrous moonshine?

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If a magic square of order n is normal (i.e., it contains the numbers 1 to n2), then the magic constant depends only on n; its value is . This formula is a consequence of the formula for the sum of the first n integers applied to the case k = n2, yielding n2(n2+1)/2, which is then divided by n because there ...

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Magic constant, Magic constant - Normal magic squares, Magic constant - External link

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Leonidas Alaoglu and Paul Erdős proved [AlaErd44] that if n is superabundant, then there exist a2, ..., ap such that and In fact, ap is nearly always 1. It can also be shown that all superabundant numbers are Harshad numbers. ...

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Superabundant number, Superabundant number - Properties

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The following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in OEIS of the sequence of k-hyperperfect numbers: It can be shown that if k > 1 is an odd integer and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then p²q is k-hyperperfect; Judson S. McCraine has conjectured in 2000 that all k-hyperperfect numbers for odd k > 1 are of this form, but the hypothesis has not been pr ...

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Hyperperfect number, Hyperperfect number - List of hyperperfect numbers

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Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. Therefore numerical calculations must use approximations of π. For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more accuracy. The approximations 22/7 and 355/113, with 3 and 7 significant figures respectively, are obtained from the simple continued fraction expansion of π. [The approximation 355/113 is the very best one that may exp ...

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Pi, Pi - Properties, Pi - Formulae involving π, Pi - Geometry, Pi - Analysis, Pi - Continued fractions, Pi - Number theory, Pi - Dynamical systems and ergodic theory, Pi - Physics, Pi - Probability and statistics, Pi - History of π, Pi - Numerical approximations of π, Pi - Miscellaneous formulae, Pi - Less accurate approximations, Pi - Open questions, Pi - The nature of π, Pi - Fictional references, Pi - π culture, Pi - Memorizing Pi

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The basic combinatorial question is how many different polyiamonds with a given number of triangles exist. If mirror images are considered identical, the number of possible n-iamonds for n = 1, 2, 3, … is (sequence A000577 in OEIS): 1, 1, 1, 3, 4, 12, 24, 66, 160, … As with polyominoes, fixed polyiamonds (where different orientations count as distinct) and one-sided polyiamonds (where mirror images count as distinct but rotations count as identical) may also be de ...

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Polyiamond, Polyiamond - Counting polyiamonds, Polyiamond - Symmetries, Polyiamond - Generalizations

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The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers (The proof of this is below). For example, we can write and any other such factorization of 23244 will be identical except for the order of the factors. See prime fac ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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Mathematically, a space group is a symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n independent directions, such as, for n = 3, a crystal. Only discrete symmetry groups are included in the categorization; i.e., infinitely fine structure or homogeneity in one or more directions is excluded. Two symmetry groups are of the same crystallographic space group type if they are the same up to an affine transformation of space that preserves orientation. Thus e.g. a change of angle between translation vectors does not affect the space group typ ...

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Space group, Space group - Space groups in crystallography, Space group - Glide planes and screw axes, Space group - Notation, Space group - Group theory, Space group - Space groups in various dimensions, Space group - Grouping space groups by point group, Space group - Further categorizing of space groups

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The integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. Another way of stating the same is that for every prime divisor p of n, the prime p does not divide n / p. Yet another formulation: n is square-free if and only if in every factorization n=ab, the factors a and b are coprime. The positive integer nSee also:

Square-free integer, Square-free integer - Equivalent characterizations of square-free numbers, Square-free integer - Distribution of square-free numbers, Square-free integer - Erdös Squarefree Conjecture

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There is also a numerical pattern discovered by Anon Castillo using 2, 3, 5, and 23 and including 17, where you add one and two, then add an increasing number to each sum: 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 (or 2^3) 4 + 8 = 12 5 + 12 = 17 6 + 17 = 23 7 + 23 = 30 (or 2 * 3 * 5) This pattern (sequence A089071 in OEIS) consists of the Triangular numbers plus 2. ...

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23 numerology, 23 numerology - A 23 enigma list, 23 numerology - 23 pattern

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One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, which means the proposition is proved by assuming that the opposite of the proposition is true and showing that this assumption is false, which means that the proposition must be true. Assume that is a rational number, meaning that there exist an integer a and an integer b such that a / b = . Then can be written as an irreducible fraction (the fraction is shortened as muc ...

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Square root of 2, Square root of 2 - History, Square root of 2 - Proof of irrationality, Square root of 2 - A different proof

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Euclid discovered that the first four perfect numbers are generated by the formula 2n−1(2n − 1): for n = 2:   21(22 − 1) = 6 for n = 3:   22(23 − 1) = 28 for n = 5:   24(25 − 1) = 496 for n = 7: ...

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Perfect number, Perfect number - Even perfect numbers, Perfect number - Odd perfect numbers

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The Pell numbers are defined recursively by: In words: you start with 0 and 1, and then produce the next Pell number by adding twice the previous Pell number to the Pell number before that. The first few terms of the sequence are (sequence A000129 in OEIS): 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... ...

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Pell number, Pell number - Pell numbers, Pell number - Pell prime, Pell number - Companion Pell numbers Pell-Lucas numbers

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The identity shows that Mn can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. The converse statement, namely that Mn is necessarily prime if n is prime, is false. The smallest counterexample is , a composite number. Fast algorithms for finding Mersenne primes are available, and this i ...

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Mersenne prime, Mersenne prime - Searching for Mersenne primes, Mersenne prime - Theorems about Mersenne prime, Mersenne prime - List of Mersenne primes

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Given a fixed oriented line L in the Euclidean plane R2, a meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane. Meander mathematics - Examples. The meander of order 1 intersects the line twice: The meanders of order 2 intersect the line four times:

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Meander mathematics, Meander mathematics - Meander, Meander mathematics - Examples, Meander mathematics - Meandric numbers, Meander mathematics - Open meander, Meander mathematics - Examples, Meander mathematics - Open meandric numbers, Meander mathematics - Semi-meander, Meander mathematics - Examples, Meander mathematics - Semi-meandric numbers, Meander mathematics - Properties of meandric numbers

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