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20-588: 48 is the 8th highly composite number , and a Størmer number . By a classical result of Honsberger , the number of incongruent integer-sided triangles of perimeter m {\displaystyle m} is given by the equations m 2 48 {\displaystyle {\tfrac {m^{2}}{48}}} for even m {\displaystyle m} , and ( m + 3 ) 2 48 {\displaystyle {\tfrac {(m+3)^{2}}{48}}} for odd m {\displaystyle m} . 48

40-549: A 0 a 1 ⋯ a n {\displaystyle a_{0}a_{1}\cdots a_{n}} . Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic , every positive integer n has a unique prime factorization: where p 1 < p 2 < ⋯ < p k {\displaystyle p_{1}<p_{2}<\cdots <p_{k}} are prime, and

60-407: A product of two numbers in 36 different ways. The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes: where a n {\displaystyle a_{n}} is the n {\displaystyle n} th successive prime number, and all omitted terms ( a 22 to a 228 ) are factors with exponent equal to one (i.e.

80-464: A similar inquiry concerning the number 5040. The first 41 highly composite numbers are listed in the table below (sequence A002182 in the OEIS ). The number of divisors is given in the column labeled d ( n ). Asterisks indicate superior highly composite numbers . The divisors of the first 19 highly composite numbers are shown below. The table below shows all 72 divisors of 10080 by writing it as

100-406: Is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers . The name "primorial", coined by Harvey Dubner , draws an analogy to primes similar to the way the name "factorial" relates to factors . For the n th prime number p n , the primorial p n #

120-481: Is a product of primorials (e.g. 360 = 2 × 6 × 30 ). Primorials are all square-free integers , and each one has more distinct prime factors than any number smaller than it. For each primorial n , the fraction ⁠ φ ( n ) / n ⁠ is smaller than for any lesser integer, where φ is the Euler totient function . Any completely multiplicative function is defined by its values at primorials, since it

140-442: Is defined as the product of the first n primes: where p k is the k th prime number. For instance, p 5 # signifies the product of the first 5 primes: The first five primorials p n # are: The sequence also includes p 0 # = 1 as empty product . Asymptotically, primorials p n # grow according to: where o ( ) is Little O notation . In general, for a positive integer n , its primorial, n# ,

160-469: Is defined by its values at primes, which can be recovered by division of adjacent values. Base systems corresponding to primorials (such as base 30, not to be confused with the primorial number system ) have a lower proportion of repeating fractions than any smaller base. Every primorial is a sparsely totient number . The n -compositorial of a composite number n is the product of all composite numbers up to and including n . The n -compositorial

180-526: Is the order of full octahedral symmetry , which describes three-dimensional mirror symmetries associated with the regular octahedron and cube . Forty-eight may also refer to: Highly composite number A highly composite number is a positive integer that has more divisors than all smaller positive integers. 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

200-447: Is the product of the primes that are not greater than n ; that is, where π ( n ) is the prime-counting function (sequence A000720 in the OEIS ), which gives the number of primes ≤ n . This is equivalent to: For example, 12# represents the product of those primes ≤ 12: Since π (12) = 5 , this can be calculated as: Consider the first 12 values of n # : We see that for composite n every term n # simply duplicates

220-622: The OEIS ) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS ). Highly composite numbers whose number of divisors is also a highly composite number are It is extremely likely that this sequence is complete. A positive integer n is a largely composite number if d ( n ) ≥ d ( m ) for all m ≤ n . The counting function Q L ( x ) of largely composite numbers satisfies for positive c and d with 0.2 ≤ c ≤ d ≤ 0.5 {\displaystyle 0.2\leq c\leq d\leq 0.5} . Because

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240-504: The infinitude of the prime numbers , where it is used to derive the existence of another prime. Notes: Primorials play a role in the search for prime numbers in additive arithmetic progressions . For instance, 2 236 133 941  + 23# results in a prime, beginning a sequence of thirteen primes found by repeatedly adding 23#, and ending with 5 136 341 251 . 23# is also the common difference in arithmetic progressions of fifteen and sixteen primes. Every highly composite number

260-536: The above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors. If Q ( x ) denotes the number of highly composite numbers less than or equal to x , then there are two constants a and b , both greater than 1, such that The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We have and Highly composite numbers greater than 6 are also abundant numbers . One need only look at

280-414: The exponents c i {\displaystyle c_{i}} are positive integers. Any factor of n must have the same or lesser multiplicity in each prime: So the number of divisors of n is: Hence, for a highly composite number n , Also, except in two special cases n  = 4 and n  = 36, the last exponent c k must equal 1. It means that 1, 4, and 36 are

300-455: 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. The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (=  7! ), as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into

320-391: The number is 2 14 × 3 9 × 5 6 × ⋯ × 1451 {\displaystyle 2^{14}\times 3^{9}\times 5^{6}\times \cdots \times 1451} ). More concisely, it is the product of seven distinct primorials: where b n {\displaystyle b_{n}} is the primorial

340-410: The only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials or, alternatively, the smallest number for its prime signature . Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 2 × 3 satisfies

360-438: The preceding term ( n − 1)# , as given in the definition. In the above example we have 12# = p 5 # = 11# since 12 is a composite number. Primorials are related to the first Chebyshev function , written ϑ ( n ) or θ ( n ) according to: Since ϑ ( n ) asymptotically approaches n for large values of n , primorials therefore grow according to: The idea of multiplying all known primes occurs in some proofs of

380-427: The prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number . Due to their ease of use in calculations involving fractions , many of these numbers are used in traditional systems of measurement and engineering designs. Primorial In mathematics , and more particularly in number theory , primorial , denoted by " p n # ",

400-487: The three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800. 10 of the first 38 highly composite numbers are superior highly composite numbers . The sequence of highly composite numbers (sequence A002182 in

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