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2520 (number)
2520 (two thousand five hundred twenty) is the natural number following 2519 and preceding 2521. In mathematics 2520 is: *the smallest number divisible by all integers from 1 to 10, i.e., it is their least common multiple. *half of 7! ( 5040), meaning 7 factorial, or 1×2×3×4×5×6×7. *the product of five consecutive numbers, namely 3×4×5×6×7. *a superior highly composite number. *a colossally abundant number. *the last highly composite number which is half of the next highly composite number. *the last highly composite number that is a divisor of all following highly composite numbers. *palindromic in bases 11 (199111), and a repdigit in bases 55, 59 and 62. *a Harshad number in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16. *the aliquot sum of 1080. *part of the 53-aliquot tree. The complete aliquot sequence starting at 1080 is: 1080, 2520, 6840, 16560, 41472, 82311, 27441, 12209, 451, 53, 1, 0. Factors The factors, also called divisor In mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal numbers'', and numbers used for ordering are called '' ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2519 (number)
2000 (two thousand) is a natural number following 1999 and preceding 2001. It is: :*the highest number expressible using only two unmodified characters in Roman numerals (MM) :*an Achilles number :*smallest four digit eban number Selected numbers in the range 2001–2999 2001 to 2099 * 2001 – sphenic number * 2002 – palindromic number * 2003 – Sophie Germain prime and the smallest prime number in the 2000s * 2004 – Area of the 24tcrystagon* 2005 – A vertically symmetric number * 2006 – number of subsets of with relatively prime elements * 2007 – 22007 + 20072 is prime * 2008 – number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to 3 * 2009 = 74 − 73 − 72 * 2010 – number of compositions of 12 into relatively prime parts * 2011 – Sexy prime with 2017, sum of eleven consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 * 2012 – The number 8 × 102012 − 1 is a prime number * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2521 (number)
2000 (two thousand) is a natural number following 1999 and preceding 2001. It is: :*the highest number expressible using only two unmodified characters in Roman numerals (MM) :*an Achilles number :*smallest four digit eban number Selected numbers in the range 2001–2999 2001 to 2099 * 2001 – sphenic number * 2002 – palindromic number * 2003 – Sophie Germain prime and the smallest prime number in the 2000s * 2004 – Area of the 24tcrystagon* 2005 – A vertically symmetric number * 2006 – number of subsets of with relatively prime elements * 2007 – 22007 + 20072 is prime * 2008 – number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to 3 * 2009 = 74 − 73 − 72 * 2010 – number of compositions of 12 into relatively prime parts * 2011 – Sexy prime with 2017, sum of eleven consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 * 2012 – The number 8 × 102012 − 1 is a prime number * 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Common Multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by both ''a'' and ''b''. Since division of integers by zero is undefined, this definition has meaning only if ''a'' and ''b'' are both different from zero. However, some authors define lcm(''a'',0) as 0 for all ''a'', since 0 is the only common multiple of ''a'' and 0. The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The least common multiple of more than two integers ''a'', ''b'', ''c'', . . . , usually denoted by lcm(''a'', ''b'', ''c'', . . .), is also well defined: It is the smallest positive integer that is divisible by each of ''a'', ''b'', ''c'', . . . Overview A multiple of a number is the product of that number and an integer. For example, 10 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5040 (number)
5040 is a factorial (7!), a superior highly composite number, abundant number, colossally abundant number and the number of permutations of 4 items out of 10 choices (10 × 9 × 8 × 7 = 5040). It is also one less than a square, making (7, 71) a Brown number pair. Philosophy Plato mentions in his ''Laws'' that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or ''polis'') into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a ''polis''. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2520 is). He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superior Highly Composite Number
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number ''n'' there exists a positive real number ''ε'' such that for all natural numbers ''k'' smaller than ''n'' we have :\frac\geq\frac and for all natural numbers ''k'' larger than ''n'' we have :\frac>\frac where ''d(n)'', the divisor function, denotes the number of divisors of ''n''. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12. \frac\approx 1.414, \frac=1.5, \frac\approx 1.633, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colossally Abundant Number
In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Formally, a number ''n'' is said to be colossally abundant if there is an ε > 0 such that for all ''k'' > 1, :\frac\geq\frac where ''σ'' denotes the sum-of-divisors function. All colossally abundant numbers are also superabundant numbers, but the converse is not true. The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 superior highly composite numbers, but neither set is a subset of the other. History Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers. Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in fina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repdigit
In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary). Repdigits are the representation in base B of the number x\frac where 0 1 and ''n'', ''m'' > 2 : **(''p'', ''x'', ''y'', ''m'', ''n'') = (31, 5, 2, 3, 5) corresponding to 31 = 111112 = 1115, and, **(''p'', ''x'', ''y'', ''m'', ''n'') = (8191, 90, 2, 3, 13) corresponding to 8191 = 11111111111112 = 11190, with 11111111111 is the repunit with thirteen digits 1. *For each sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harshad Number
In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit ' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. Definition Stated mathematically, let be a positive integer with digits when written in base , and let the digits be a_i (i = 0, 1, \ldots, m-1). (It follows that a_i must be either zero or a positive integer up to .) can be expressed as :X=\sum_^ a_i n^i. is a harshad number in base if: :X \equiv 0 \bmod . A number which is a harshad number in every number base is called an all-harshad number, or an all-Niven number. There are only four all-harshad numbers: 1, 2, 4, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sum
In number theory, the aliquot sum ''s''(''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, :s(n)=\sum\nolimits_d. It can be used to characterize the prime numbers, perfect numbers, " sociable numbers", deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. Examples For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6). The values of ''s''(''n'') for ''n'' = 1, 2, 3, ... are: :0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... Characterization of classes of numbers The aliquot sum function can be used to characterize several notable classes of numbers: *1 is the only number whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sequence
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. Definition and overview The aliquot sequence starting with a positive integer ''k'' can be defined formally in terms of the sum-of-divisors function σ1 or the aliquot sum function ''s'' in the following way: : ''s''0 = ''k'' : ''s''n = ''s''(''s''''n''−1) = σ1(''s''''n''−1) − ''s''''n''−1 if ''s''''n''−1 > 0 : ''s''n = 0 if ''s''''n''−1 = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all aliquot sequences would be infinite sequence, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6) and ''s''(0) is undefined. For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because: :σ1(10) − 10 = 5 + 2 + 1 = 8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
