45 ( forty-five ) is the natural number following 4
13-563: 4 and preceding 46 . Forty-five is the smallest odd number that has more divisors than n + 1 {\displaystyle n+1} , and that has a larger sum of divisors than n + 1 {\displaystyle n+1} . It is the sixth positive integer with a square-prime prime factorization of the form p 2 q {\displaystyle p^{2}q} , with p {\displaystyle p} and q {\displaystyle q} prime . 45 has an aliquot sum of 33 that
26-583: A total of 45 classes of finite simple groups. Forty-four is: AD 44 , 44 BC , 1944 , etc. St%C3%B8rmer number In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n {\displaystyle n} for which the greatest prime factor of n 2 + 1 {\displaystyle n^{2}+1} is greater than or equal to 2 n {\displaystyle 2n} . They are named after Carl Størmer . The first few Størmer numbers are: John Todd proved that this sequence
39-488: Is half of a right angle (90°). In the classification of finite simple groups , the Tits group T {\displaystyle \mathbb {T} } is sometimes defined as a nonstrict group of Lie type or sporadic group , which yields a total of 45 classes of finite simple groups : two stem from cyclic and alternating groups , sixteen are families of groups of Lie type, twenty-six are strictly sporadic, and one
52-456: Is neither finite nor cofinite . More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905. It has been conjectured that their natural density is the natural logarithm of 2 , approximately 0.693, but this remains unproven. Because the Størmer numbers have positive density, the Størmer numbers form a large set . The Størmer numbers arise in connection with
65-476: Is part of an aliquot sequence composed of five composite numbers (45, 33, 15 , 9 , 4 , 3 , 1 , and 0 ), all of which are rooted in the 3 -aliquot tree. This is the longest aliquot sequence for an odd number up to 45. Forty-five is the sum of all single-digit decimal digits: 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 {\displaystyle 0+1+2+3+4+5+6+7+8+9=45} . It is, equivalently,
78-449: Is the exceptional case of T {\displaystyle \mathbb {T} } . Forty-five may also refer to: 44 (number) 44 ( forty-four ) is the natural number following 43 and preceding 45 . Forty-four is a repdigit and palindromic number in decimal . It is the tenth 10- happy number , and the fourth octahedral number . It is a square-prime of the form p × q , and fourth of this form and of
91-499: The Gaussian integer a + b i {\displaystyle a+bi} by numbers of the form n ± i {\displaystyle n\pm i} , in order to cancel prime factors p {\displaystyle p} from the imaginary part; here n {\displaystyle n} is chosen to be a Størmer number such that n 2 + 1 {\displaystyle n^{2}+1}
104-433: The truncated cube and truncated octahedron , per Miller's rules . 44 four-dimensional crystallographic point groups of a total 227 contain dual enantiomorphs , or mirror images. There are forty-four classes of finite simple groups that arise from four general families of such groups: Sometimes the Tits group is considered a 17th non-strict simple group of Lie type, or a 27th sporadic group, which would yield
117-408: The form 2 × q , where q is a higher prime. It is the first member of the first cluster of two square-primes; of the form p × q , specifically 2 × 11 = 44 and 3 × 5 = 45 . The next such cluster of two square-primes comprises 2 × 29 = 116 , and 3 × 13 = 117 . 44 has an aliquot sum of 40 , within an aliquot sequence of three composite numbers (44, 40, 50 , 43 , 1 , 0 ) rooted in
130-674: The greatest prime factor of 45 2 + 1 = 2026 {\displaystyle 45^{2}+1=2026} is 1,013, which is much more than 45 twice, 45 is a Størmer number . In decimal, 45 is a Kaprekar number and a Harshad number . Forty-five is a little Schroeder number ; the next such number is 197 , which is the 45th prime number. Forty-five is conjectured from Ramsey number R ( 5 , 5 ) {\displaystyle R(5,5)} . ϕ ( 45 ) = ϕ ( σ ( 45 ) ) {\displaystyle \phi (45)=\phi (\sigma (45))} Forty-five degrees
143-711: The ninth triangle number . Forty-five is also the fourth hexagonal number and the second hexadecagonal number , or 16-gonal number. It is also the second smallest triangle number (after 1 and 10) that can be written as the sum of two squares. Forty-five is the smallest positive number that can be expressed as the difference of two nonzero squares in more than two ways: 7 2 − 2 2 {\displaystyle 7^{2}-2^{2}} , 9 2 − 6 2 {\displaystyle 9^{2}-6^{2}} or 23 2 − 22 2 {\displaystyle 23^{2}-22^{2}} (see image). Since
SECTION 10
#1732776631349156-519: The prime 43 -aliquot tree. Since the greatest prime factor of 44 + 1 = 1937 is 149 and thus more than 44 twice, 44 is a Størmer number . Given Euler's totient function , φ(44) = 20 and φ(69) = 44. 44 is a tribonacci number , preceded by 7, 13, and 24, whose sum it equals. 44 is the number of derangements of 5 items. There are only 44 kinds of Schwarz triangles , aside from the infinite dihedral family of triangles ( p 2 2) with p = {2, 3, 4, ...}. There are 44 distinct stellations of
169-507: The problem of representing the Gregory numbers ( arctangents of rational numbers ) G a / b = arctan b a {\displaystyle G_{a/b}=\arctan {\frac {b}{a}}} as sums of Gregory numbers for integers (arctangents of unit fractions ). The Gregory number G a / b {\displaystyle G_{a/b}} may be decomposed by repeatedly multiplying
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