This article may require cleanup to meet Wikipedia's quality standards. The specific problem is: Explanatory footnotes are about twice the length of article prose. Move content to main article prose, linked articles, or reduce level of detail. People using screen readers hear explanatory footnotes at the end of the article, meaning they lose context unless there is a small enough number to keep track of mentally.(March 2024) |
| ||||
---|---|---|---|---|
Cardinal | seven hundred forty-four | |||
Ordinal | 744th (seven hundred forty-fourth) | |||
Factorization | 23 × 3 × 31 | |||
Divisors | 1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744 | |||
Greek numeral | ΨΜΔ´ | |||
Roman numeral | DCCXLIV | |||
Binary | 10111010002 | |||
Ternary | 10001203 | |||
Senary | 32406 | |||
Octal | 13508 | |||
Duodecimal | 52012 | |||
Hexadecimal | 2E816 |
744 (seven hundred [and] forty four) is the natural number following 743 and preceding 745.
744 plays a major role within moonshine theory of sporadic groups, in context of the classification of finite simple groups.
744 is the nineteenth number of the form where , and represent distinct prime numbers (2, 3, and 31; respectively). [1]
It can be represented as the sum of nonconsecutive factorials , [2] as the sum of four consecutive primes , [3] and as the product of sums of divisors of consecutive integers ; [4] respectively: [lower-roman 1]
744 contains sixteen total divisors — fourteen aside from its largest and smallest unitary divisors — all of which collectively generate an integer arithmetic mean of [10] [11] that is also the first number of the form [1] [lower-roman 2]
The number partitions of the square of seven (49) into prime parts is 744, [17] as is the number of partitions of 48 into at most four distinct parts. [18] [lower-roman 3]
744 is an abundant number, [23] with an abundance of 432. [24] [lower-roman 4] It is semiperfect, since it is equal to the sum of a subset of its divisors (e.g., 1 + 2 + 4 + 24 + 62 + 93 + 124 + 186 + 248). [33] [lower-roman 5]
It is also a practical number, [50] and the first number to be the sum of nine cubes in eight or more ways, [51] as well as the number of six-digit perfect powers in decimal. [52]
Meanwhile, in septenary 744 is palindromic (21127), [53] while in binary it is a pernicious number, as its digit representation (10111010002) contains a prime count (5) of ones. [54] [lower-roman 7]
744 has two hundred and forty integers that are relatively prime or coprime with and up to itself, equivalently its Euler totient. [5] [lower-roman 8]
This totient of 744 is regular like its sum-of-divisors, where 744 sets the twenty-ninth record for of 1920. [66] [lower-roman 9] Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors (2, 3, and 5), [68] while the Carmichael function or reduced totient (which counts the least common multiple of order of elements in a multiplicative group of integers modulo ) at seven hundred forty-four is equal to . [26] [lower-roman 10] 744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum (960). [80] [lower-roman 11]
Of these 240 totatives, 110 are strictly composite totatives that nearly match the sequence of composite numbers up to 744 that are congruent to , which is the same congruence that all prime numbers greater than 3 hold. [88] [lower-roman 12] Only seven numbers present in this sequence are not totatives of 744 (less-than); they are 713, 589, 527, 403, 341, 217, and 155; all of which are divisible by the eleventh prime number 31. [lower-roman 13] [lower-roman 14] [lower-roman 15] The remaining 130 totatives are 1 and all the primes between 5 and 743 except for 31 (all prime numbers less than 744 that are not part of its prime factorization) where its largest prime totative of 743 [lower-roman 16] has a prime index of 132 (the smallest digit-reassembly number in decimal). [127] [lower-roman 17] On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238. [5] [lower-roman 18]
744 is the sixth number whose totient value has a sum-of-divisors equal to : . [131] Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176 [14] which is the forty-eighth triangular number, [6] and the binomial coefficient present inside the forty-ninth row of Pascal's triangle. [132] [lower-roman 19] In total, only seven numbers have sums of divisors equal to 744; they are: 240, [lower-roman 20] 350, 366, 368, 575, 671, and 743. [32] [lower-roman 21] When only the fourteen proper divisors of 744 are considered, then the sum generated by these is 1175, whose six divisors contain an arithmetic mean of 248, [11] [lower-roman 22] the third (or fourteenth) largest divisor of 744. Only one number has an aliquot sum that is 744, it is 456. [14] [lower-roman 23]
The number of Euler tours (or Eulerian cycles) of the complete, undirected graph on six vertices and fifteen edges is 744. [164] On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and 264 (the latter is the second digit-reassembly number in base ten). [127] On the other hand, the number of Euler tours of the complete digraph, or directed graph, on four vertices is 256, while on five vertices it is 972,000 (and 247,669,456,896 on six vertices), by the BEST theorem. [165]
Regarding the largest prime totative of 744, there are (aside from the sets that are the union of all such solutions),
Thrice 743 is 2229, [lower-roman 24] whose average of divisors is 744 (as with thrice any prime number , the average of divisors will be ). [10] [11] [lower-roman 25] This value is a difference of 1110 from 3339, which is the sum of seven hundred and forty-two ( 742 ) [lower-roman 26] repeating digits of the reciprocal of 743, as the forty-eighth full repetend prime in decimal [25] (with the smallest number to have a Euler totient of 744 being 1119). [5]
For open trails of lengths eight and nine, starting and ending at fixed distinct vertices in the complete undirected graph on five labeled vertices, the number is 132 (the prime index of 743, half 264), [195] that also represents the number of irreducible trees with fifteen vertices. [196] [lower-roman 27] While for the complete undirected graph there are 264 directed Eulerian circuits, [201] [202] it is more specifically the number of circuits of length ten in the complete undirected graph on five labeled vertices, and as such it is the twenty-fifth element in a triangle of length on labeled vertices. [203]
Otherwise, 745 is the number of disconnected simple labeled graphs covering six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges intersecting; this yields the disconnected covering graph on vertices labelled through in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations. [204]
456 (the only number to have an aliquot sum of 744) is the number of unlabeled non-mating graphs with seven vertices (where a mating or graph is a graph where no two vertices have the same set of neighbors), equivalently the number of unlabeled graphs with seven vertices and at least one endpoint; [205] as well as the number of cliques in the 7-triangular graph, where every subset of two distinct vertices in a clique are adjacent. [206] The number of even graphs with seven vertices, where a graph is odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise, is 456. [207] [208] [lower-roman 28]
In particular, 456 is the aliquot sum of 264, the only number to have this value for [14] [lower-roman 29]
744 is the twelfth self-convolution of Fibonacci numbers, which is equivalently the number of elements in all subsets of with no consecutive integers. [225] [226] [227]
The j–invariant holds as a Fourier series q–expansion,
where and the half-period ratio of an elliptic function. [228]
The j–invariant can be computed using Eisenstein series and , such that:
where specifically, these Eisenstein series are equal to and with .
The respective q–expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively; [229] [230] where specifically the sum and difference between the absolute values of these two numbers is 240 + 504 = 744 and 504 − 240 = 264. Furthermore, when considering the only smaller even (here, non-modular) series the sum between the absolute value of its constant multiplicative term (24) and that of (240) is equal to 264 as well. The 16th coefficient in the expansion of is −744, as is its 25th coefficient. [231]
Alternatively, the j–invariant can be computed using a sextic polynomial as: where represents the modular function, with 256 = 28. [232]
Also, the almost integer [233]
This number is known as Ramanujan's constant, which is transcendental. [234] Mark Ronan and other prominent mathematicians have noted that the appearance of in this number is relevant within moonshine theory , where one hundred and sixty-three is the largest of nine Heegner numbers that are square-free positive integers such that the imaginary quadratic field has class number of (equivalently, the ring of integers over the same algebraic number field have unique factorization). [235] : pp.227, 228 has one hundred and ninety-four (194 = 97 × 2) conjugacy classes generated from its character table that collectively produces the same number of elliptic moonshine functions which are not all linearly independent; only one hundred and sixty-three are entirely independent of one another. [236] The linear term of error for Ramanujan's constant is approximately,
where is the value of the minimal faithful complex dimensional representation of the Friendly Giant , the largest sporadic group. [237] contains an infinitely graded faithful dimensional representation equivalent to the coefficients of the series of the q-expansion of the j-invariant. Specifically, all three common prime factors that divide the Euler totient, sum-of-divisors, and reduced totient of are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, ; [235] : pp.244–246 [lower-greek 1] three of these belong inside the small family of six pariah groups that are not subquotients of [245] The largest supersingular prime that divides the order of is [246] [247] which is the eighth self-convolution of Fibonacci numbers, where is the twelfth. [226] [lower-greek 2]
The largest three Heegner numbers with also give rise to almost integers of the form which involve . In increasing orders of approximation, [263] : p.20–23 [lower-greek 3]
Square-free positive integers over the negated imaginary quadratic field with class number of also produce almost integers for values of , where for instance there is [266] [267] [lower-greek 4] [lower-greek 5]
is theta series coefficient of four-dimensional cubic lattice [280] [281] On the other hand, in the theta series of the four-dimensional body-centered cubic lattice — whose geometry with defines Hurwitz quaternions of even and odd square norm as realized in the –cell honeycomb that is dual to the –cell honeycomb (and, as a union of two self-dual tesseractic –cell honeycombs) — the sixteenth coefficient is the seven hundred and forty-fourth coefficient in the series; [282] with coefficient index the forty-ninth non-zero norm. [283]
4 +≅ as a lattice is the union of two lattices, [281] where the associated theta series of has 744 as its 50th indexed coefficient (as with theta series of ), [284] [280] and where its twenty-fifth coefficient is 248, which is the most important divisor of 744. Also, in this theta series of , the preceding 49th coefficient is 456, the only number to hold an aliquot sum of 744, [14] where 744 − 456 = 288 , a value that is the number of cells in the disphenoidal 288-cell, whose 48 vertices collectively represent the twenty-four Hurwitz unit quaternions with norm squared 1, united with the twenty-four vertices of the dual 24-cell with norm squared 2. For the theta series of , that is realized in the 16-cell honeycomb, all 25 × 2m indexed coefficients (i.e. 25, 50 , 100, 200, 400, ...) are 744. [284] For both the theta series of and , the 456th coefficient is 1920, the sum-of-divisors of 744. [32]
For the theta series of the four-dimensional lattice, coefficient index 288 is the 97th non-zero norm, coeff. index 456 the 153rd (an index that represents the seventeenth triangular number), and coeff. index 744 the 250th; [283] the latter, a coefficient index that is the largest of only four numbers to hold a Euler totient of 100: 101, 125, 202, and 250 [5] — the smallest of these is the twenty-sixth prime number, while the largest 250 = 2 × 53 has a sum of prime factors that is 17, the seventh prime number; [9] and where 202 = 49 + 153 and 250 = 153 + 97, with 549 = 49 + 97 + 153 + 250, the 447th indexed composite number. [28] More deeply, for the theta series of 744 is a prime-indexed coefficient over its first six indices less than 104 (respectively, the 458th, 526th, 742nd, 799th, 842nd, and 1141st prime indices, with a sum of 4058; the latter in equivalence with 1141 = 7 × 163 ). [282] The first composite coeff. index in the series with coefficient 744 is its seventh index 9251 = 11 × 292, whose sum of prime factors, inclusive of 1, is 70 (which is of algebraic significance in terms of the construction of the twenty-four-dimensional Leech lattice). 456, and 1176, the aliquot sum of 744, [14] are also prime-indexed coefficients over their first two coefficient indices (346th, 364th, and 1098th, 1159th, respectively). [282]
In four-dimensional space, three-dimensional cell facets of the three-largest of six regular –polytopes (the octaplex, dodecaplex, and tetraplex) [285] : p.292 collectively number [lower-greek 6]
Within finite simple groups of Lie type, exceptional Lie algebra holds a minimal faithful representation in two hundred and forty-eight dimensions, where divides thrice over. [286] [287] : p.4 John McKay noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the Dynkin diagrams of complex Lie algebra as well as those of and respectively coincide with the three largest conjugacy classes of ; where also the corresponding McKay–Thompson series of sporadic Thompson group holds coefficients representative of its faithful dimensional representation (also minimal at ) [288] [237] whose values themselves embed irreducible representation of . [289] : p.6 In turn, exceptional Lie algebra is shown to have a graded dimension [290] whose character lends to a direct sum equivalent to, [289] : p.7, 9–11
The twenty-four dimensional Leech lattice in turn can be constructed using three copies of the associated lattice [292] [285] : pp.233–235 [lower-greek 7] and with the eight-dimensional octonions (see also, Freudenthal magic square), [297] where the automorphism group of is the smallest exceptional Lie algebra , which embeds inside . In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from (as ) with a central charge of , out of a total seventy-one such modular invariant conformal field theories of holomorphic VOAs of weight one. [298] Known as Schellekens' list, these algebras form deep holes in whose corresponding orbifold constructions are isomorphic to the moonshine module ♮ that contains as its automorphism; [299] of these, the second and third largest contain affine structures and that are realized in . [lower-greek 8] [304] [305]
is also the sum of consecutive pentagonal numbers, [306] [307] [lower-roman 34]
It is a "selfie number", where [309] [310] [lower-roman 35]
is the magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between and inclusive. [311] [lower-roman 36]
It is the number of non-congruent polygonal regions in a regular –gon with all diagonals drawn. [319] [lower-roman 37]
There are seven hundred and forty-four ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle. [321] [lower-roman 38]
10 (ten) is the even natural number following 9 and preceding 11. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
111 is the natural number following 110 and preceding 112.
90 (ninety) is the natural number following 89 and preceding 91.
27 is the natural number following 26 and preceding 28.
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or 6 dozen.
32 (thirty-two) is the natural number following 31 and preceding 33.
34 (thirty-four) is the natural number following 33 and preceding 35.
55 (fifty-five) is the natural number following 54 and preceding 56.
58 (fifty-eight) is the natural number following 57 and preceding 59.
63 (sixty-three) is the natural number following 62 and preceding 64.
64 (sixty-four) is the natural number following 63 and preceding 65.
104 is the natural number following 103 and preceding 105.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
360 is the natural number following 359 and preceding 361.
144 is the natural number following 143 and preceding 145.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
168 is the natural number following 167 and preceding 169.
177 is the natural number following 176 and preceding 178.
1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.
888 is the natural number following 887 and preceding 889.
{{cite journal}}
: CS1 maint: Zbl (link)n | |nint(x) − x| |
---|---|
25 | −0.00066 |
37 | −0.000022 |
43 | −0.00022 |
58 | −1.8×10−7 |
67 | −1.3×10−6 |
74 | −0.00083 |
148 | 0.00097 |
163 | −7.5×10−13 |
232 | −7.8×10−6 |
268 | 0.00029 |
522 | −0.00015 |
652 | 1.6×10−10 |
719 | −0.000013 |
{{cite book}}
: CS1 maint: location missing publisher (link){{cite journal}}
: CS1 maint: Zbl (link)