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Table Of Divisors
The tables below list all of the divisors of the numbers 1 to 1000. A divisor of an integer ''n'' is an integer ''m'', for which \frac is again an integer (which is necessarily also a divisor of ''n''). For example, 3 is a divisor of 21, since = 3 (and therefore 7 is also a divisor of 21). If ''m'' is a divisor of ''n'', then so is −''m''. The tables below only list positive divisors. Key to the tables * ''d''(''n'') is the number of the positive divisors of ''n'', including 1 and ''n'' itself * σ(''n'') is the sum of the positive divisors of ''n'', including 1 and ''n'' itself * ''s''(''n'') is the sum of the proper divisors of ''n'', including 1 but not ''n'' itself; that is, ''s''(''n'') = σ(''n'') − ''n'' *a deficient number is greater than the sum of its proper divisors; that is, ''s''(''n'') ''n'' *a highly abundant number has a sum of positive divisors that is greater than any lesser number; that is, σ(''n'') > σ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Numbers
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. 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. Ramanujan wrote a paper on highly composite numbers in 1915. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Abundant Number
In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any is at least proportional to . Formal definition and examples Formally, a natural number ''n'' is called highly abundant if and only if for all natural numbers ''m'' < ''n'', :\sigma(n) > \sigma(m) where σ denotes the sum-of-divisors function. The first few highly abundant numbers are : 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
9 (number)
9 (nine) is the natural number following and preceding . Evolution of the Hindu–Arabic digit Circa 300 BC, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a -look-alike. How the numbers got to their Gupta form is open to considerable debate. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase ''a''. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic. While the shape of the glyph for the digit 9 has an ascender in most modern typef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8 (number)
8 (eight) is the natural number following 7 and preceding 9. Etymology English ''eight'', from Old English '', æhta'', Proto-Germanic ''*ahto'' is a direct continuation of Proto-Indo-European '' *oḱtṓ(w)-'', and as such cognate with Greek and Latin , both of which stems are reflected by the English prefix oct(o)-, as in the ordinal adjective ''octaval'' or ''octavary'', the distributive adjective is '' octonary''. The adjective ''octuple'' (Latin ) may also be used as a noun, meaning "a set of eight items"; the diminutive '' octuplet'' is mostly used to refer to eight siblings delivered in one birth. The Semitic numeral is based on a root ''*θmn-'', whence Akkadian ''smn-'', Arabic ''ṯmn-'', Hebrew ''šmn-'' etc. The Chinese numeral, written ( Mandarin: ''bā''; Cantonese: ''baat''), is from Old Chinese ''*priāt-'', ultimately from Sino-Tibetan ''b-r-gyat'' or ''b-g-ryat'' which also yielded Tibetan '' brgyat''. It has been argued that, as the cardinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
7 (number)
7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube. As an early prime number in the series of positive integers, the number seven has symbolic associations in religion, mythology, superstition and philosophy. The seven classical planets resulted in seven being the number of days in a week. 7 is often considered lucky in Western culture and is often seen as highly symbolic. Evolution of the Arabic digit For early Brahmi numerals, 7 was written more or less in one stroke as a curve that looks like an uppercase vertically inverted (ᒉ). The western Arab peoples' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the digit more rectilinear. The eastern Arab peoples developed the digit from a form that looked something like 6 to one that looked like an uppercase V. Both modern Arab forms influenced the European form, a two-stroke form consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28. The first four perfect numbers are 6 (number), 6, 28 (number), 28, 496 (number), 496 and 8128 (number), 8128. The sum of proper divisors of a number is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. This definition is ancient, appearing as early as Euclid's Elements, Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby \frac is an even perfect number whenever q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
6 (number)
6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon also has 6 edges as well as 6 internal and external angles. 6 is the second smallest composite number. It is also the first number that is the sum of its proper divisors, making it the smallest perfect number. It is also the only perfect number that doesn't have a digital root of 1. 6 is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist. 6 is the largest of the four all-Harshad numbers. 6 is the 2nd superior highly composite number, the 2nd colossally abundant number, the 3rd triangular number, the 4th highly composite number, a pronic number, a congruent number, a harmonic divisor number, and a semiprime. 6 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 (number)
5 (five) is a number, numeral and digit. It is the natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ..., and cardinal number, following 4 and preceding 6, and is a prime number. Humans, and many other animals, have 5 Digit (anatomy), digits on their Limb (anatomy), limbs. Mathematics 5 is a Fermat prime, a Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). 5 is the first safe prime and the first good prime. 11 forms the first pair of sexy primes with 5. 5 is the second Fermat number, Fermat prime, of a total of five known Fermat primes. 5 is also the first of three known Wilso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime number, prime, or the Unit (ring theory), unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4 (number)
4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures. Evolution of the Hindu-Arabic digit Brahmic numerals represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross. While the shape of the character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3 (number)
3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious and cultural significance in many societies. Evolution of the Arabic digit The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and Chinese numerals) that are still in use. That was also the original representation of 3 in the Brahmic (Indian) numerical notation, its earliest forms aligned vertically. However, during the Gupta Empire the sign was modified by the addition of a curve on each line. The Nāgarī script rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a with an additional stroke at the bottom: ३. The Indian digits spread to the Caliphate in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superior Highly Composite Number
In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the greatest ratio is a superior highly composite 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 ten superior highly composite numbers and their factorization are listed. For a superior highly composite number there exists a positive real number such that for all natural numbers we have \frac\geq\frac where , the divisor function, denotes the number of divisors of . 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 composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
