744 (number)

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.

Number theory
744 is the nineteenth number of the form $$pqr^{3}$$ where $$r$$, $$p$$ and $$q$$ represent distinct prime numbers (2, 3, and 31; respectively).

It can be represented as the sum of nonconsecutive factorials $$k!$$, as the sum of four consecutive primes $$p$$, and as the product of sums of divisors $$\sigma(n)$$ of consecutive integers $$n$$; respectively:

$$ \begin{align} 744 & = 4! + 6! \\ 744 & = 179 + 181 + 191 + 193 \\ 744 & = \sigma(15) \times \sigma(16) = 24 \times 31 \\ \end{align}$$

744 contains sixteen total divisors — fourteen aside from its largest and smallest unitary divisors — all of which collectively generate an integer arithmetic mean of $$120 = 5!$$ that is also the first number of the form $$pqr^{3}.$$ 120 is also equal to the sum of the first fifteen integers, or fifteenth triangular number $Σ_{15} _{n = 1} n$, while it is also the smallest number with sixteen divisors, with 744 the thirty-first such number. This value is also equal to the sum of all the prime numbers less than 31 that are not factors of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to $125 = 5^{3}$, which is the second number after 32 to have an aliquot sum of 31. In the Collatz conjecture, 744 and 120 both require fifteen steps to reach 5, before cycling through {16, 8, 4, 2, 1} in five steps. Otherwise, they both require nineteen steps to reach 2, which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself, or twenty steps to reach 1.

The number partitions of the square of 7|seven (49) into prime parts is 744, as is the number of partitions of 48 into at most four distinct parts. The radical $186 = 2 × 3 × 31$ of 744 has an arithmetic mean of divisors equal to 48, where the sum between the three distinct prime factors of 744 is $6^{2} = 36$. 744 is also a practical number, and the first number to be the sum of nine cubes in eight or more ways, as well as the number of six-digit perfect powers in decimal.

φ(n) and σ(n)
744 has two hundred and forty integers that are relatively prime or coprime with and up to itself, equivalently its Euler totient $$\varphi(n)$$. 744 is the twenty-third of thirty-one such numbers to have a totient of 240, after 738, and preceding 770. The smallest is 241, the fifty-third prime number and sixteenth super-prime, and the largest is 1050, which represents the number of parts in all partitions of 29 into distinct parts.

This totient of 744 is regular like its sum-of-divisors $$\sigma(n)$$, where 744 sets the twenty-ninth record for $$\sigma(n),$$ of 1920. The value of this sigma function represents the fifteenth sum of non-triangular numbers in-between triangular numbers; in this instance it is the sum that lies in-between the fifteenth (120) and sixteenth (136) triangular numbers (i.e. the sum of $121 + 122 + ... + 135$). Both the totient and sum-of-divisors values of 744 contain the same set of distinct prime factors (2, 3, and 5), while the Carmichael function or reduced totient (which counts the least common multiple of order of elements in a multiplicative group of integers modulo $$n$$) at seven hundred forty-four is equal to $$\lambda(744) = 30 = 2 \times 3 \times 5$$.

Totatives
Of these 240 totatives, 110 are strictly composite totatives that nearly match the sequence of composite numbers up to 744 that are congruent to $$\pm {1} \bmod {6}$$, which is the same congruence that all prime numbers greater than 3 hold. 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. 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 has a prime index of 132 (the smallest digit-reassembly number in decimal). On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238.

Zumkeller number
744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum: 960.

It is the 168th indexed Zumkeller number. The two sets of divisors of 744 with equal sums are:

The index of 744 as a Zumkeller number (168) represents the product of the first two perfect numbers $465 = 3 × 5 × 31$. It is the fifth Dedekind number, where the previous four members (2, 3, 6, 20) add to 31 — with 496 the thirty-first triangular number, and third perfect number (168 is also the number of primes below 1000).

960 is the thirty-first Jordan–Pólya number that is the product of factorials $m$, equal to the sum of six consecutive prime numbers $6m + 1$, between the 35th and 40th primes (it is the thirty-fifth such number). The fifteenth and sixteenth triangular numbers generate the sum $6m − 1$ that is the totient value of 960, and the number of partitions of $(31,2)$ into distinct parts and odd parts. Like its twin prime 31, 29 is a primorial prime, which together comprise the third and largest of three known pairs of twin primes (1st, 2nd, and 5th) to be primorial primes, with possibly no larger pairs. In between these lies 30.

φ(σ(744)) = 744
744 is the sixth number $$n$$ whose totient value has a sum-of-divisors equal to $$n$$. Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176 which is the forty-eighth triangular number, and the binomial coefficient $$\operatorname {C(49,2)}$$ present inside the forty-ninth row of Pascal's triangle. 1176 is also one of two middle terms in the twelfth row of a $$\operatorname {(2,3)}-$$Pascal triangle. In the triangle of Narayana numbers, 1176 appears as the fortieth and forty-second terms in the eighth row, which also includes 336 (the totient of 1176) and 36 (the square of 6). Inside the triangle of Lah numbers of the form $$\textstyle {L(n,k) = \left\lfloor {n \atop k} \right\rfloor}$$, 1176 is a member with ${0, 1, ..., 31}$ and $2 mod 3$. It is a self-Fibonacci number; the fifty-first indexed member where in its case $$F_{1176}$$ divides $$F_{F_{1176}}$$, and the forty-first 6-almost prime that is divisible by exactly six primes with multiplicity.

In total, only seven numbers have sums of divisors equal to 744; they are: 240, 350, 366, 368, 575, 671, and 743. The sum of all these seven integers whose sum-of-divisors are in equivalence with 744 is 3313, the 466th prime number and 31st balanced prime, as the middle member of the 49th triplet of sexy primes $(28 + 6)$; it is respectively the 178th and 179th such prime $34$ where $42 + 66 = 108$ and $$\mathbb{N}$^{+} ∪ {0} = $\mathbb{N}$_{0}$ are prime — where 178 is the 132nd composite number, itself the prime index of the largest number (743) to hold a sum-of-divisors of 744. 3313 is also the 24th centered dodecagonal or star number (and 15th that is prime). Divided into two numbers, 3313 is the sum of 1656 and 1657, the latter being the 260th prime number, an index value in-turn that is the first of five numbers to have a sum-of-divisors of 588, which is half of 1176 (the aliquot sum of 744); its arithmetic mean of divisors (of 260), on the other hand, is equal to 49. Furthermore, 260 is the average of divisors of 2232 (the fourth largest of five numbers to hold this value), which is thrice 744.

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, the third (or fourteenth) largest divisor of 744. Only one number has an aliquot sum that is 744, it is 456.

Pernicious number
In binary 744 is a pernicious number, as its digit representation (10111010002) contains a prime count (5) of ones.

744 is the four hundred and sixth indexed pernicious number, where 406 is the twenty-eighth triangular number; in its base-two representation, the digit positions of zeroes are in $M(n)$ or $M(n)$ ratio with the positions of ones, which are in $20 + 57 + 86 = 163$ ratio. Its ones' complement is 1000101112, equivalent to $20 + 57 = 77$ in decimal, which represents the sum of GCDs of parts in all partitions of $e^{\pi√163}$. It is also the number of partitions of $5^{2} = 25$ (a divisor of 744) as well as 63 into factorial parts (without including 0!), and the number of integer partitions of 44 whose length is equal to the LCM of all parts (with 63 the forty-fourth composite number, where 44 is itself the number of derangements of 5, and $16 = 4^{2}$ the twenty-eighth prime number).

211210 is the number of repeating decimal digits of 2113 as a full repetend prime in decimal, where 2112 is the 65th interprime to lie between consecutive twin primes, otherwise it is the 317th member between consecutive odd primes, where 317 is the 66th indexed prime number (with 90 and 91 the 65th and 66th composites, respectively, where $p$). 279, on the other hand, is the 58th number to lie between consecutive odd prime numbers (277, 281), with $6m + 1$ the fifty-eighth composite number. More specifically, eighty-one is the sum of the repeating digits of the third full repetend prime in decimal, 19, where the sum of these digits is the magic constant of an $m$ non-normal yet full prime reciprocal magic square based on its reciprocal ($0$).

Meanwhile, in septenary 744 is palindromic (21127),

Semiperfect number
744 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).

744 is the 183rd semiperfect number, an index value whose sum-of-divisors is 248, with an arithmetic mean of divisors equal to $23 × 31$, both of which are divisors of 744 (fourteenth  and tenth largest, respectively); otherwise, the sum of its two divisors greater than 1 is $6m − 1$. 183 is also the eighth perfect totient number, and the number of semiorders on four labeled elements. It is the largest number of interior regions formed by fourteen intersecting circles, which is equivalent to the number of points in the projective plane over the finite field $$\mathbb{Z}_{13}$$, since $19 × 31$. It is the number of toothpicks in the toothpick sequence after eighteen stages. Following, 181, 183 is the sixty-second Löschian number of the form $121 = 11^{2}$, as a product of the third and twenty-fourth such numbers (3, 61), of prime indices two and eighteen. Furthermore, the smallest number to have exactly four solutions to this quadratic polynomial is the first taxicab number 1729, from $17 × 31$ integer pairs $12^{2} = 144$ and $2 + 3 + ... + 32$, where these four pairs collectively generate a sum of 183. The smallest such number with only two solutions, on the other hand, is 49; wherein 1729 is the 97th such number expressible in two, or more, ways. The probability that any odd number is a Löschian number is 0.75, while the probability that it is an even number is a remaining 0.25.

There are precisely 816 integers (uninclusively, equal to the sum-of-divisors of 737, the largest composite totative of 744) between the semiperfect index of 744 and 1000, a number that in-turn has a sum-of-divisors of $13 × 31$ — the 24th decagonal number, as well as the 30th number that can be expressed as the difference of the squares of consecutive primes in just one distinct way; other members in this sequence include: 888 is the 13th member, the 26th repdigit in decimal equal to the sum between 432 and 456, and between 144 and 744 (where 432 and 456 both have totient values of 144); and equal to the product between the twelfth prime number 37, and 24. The first four members (5, 16, 24, 48) generate a sum of 93, which is the eleventh largest divisor of 744, following 62. 93 represents the number of different cyclic Gilbreath permutations on 11 elements, and consequently there are ninety-three different real periodic points of order 11 on the Mandelbrot set.

Otherwise, $693 = 144 + 549$, in equivalence with the aliquot sum of 744 (1176) and 456, is the sixteenth number that can be expressed as the difference of the squares of primes in just two distinct ways, consecutive or otherwise (432, 1728, 1920 and 2232 are also in this sequence).

183 is also the first non-trivial 62-gonal number.

Abundance and in full repetend primes
744 is an abundant number, with an abundance of 432.

744 is the 181st abundant number. Specifically, 432 (twice 216, the cube of 6) has an aliquot sum of 808, where 433 is the thirty-first full repetend prime in decimal that repeats 432 digits, which collectively add to 1944. Less than 1000, the sixtieth and largest such prime number is 983, which repeats 982 digits that sum to 4419, a two-digit dual permutation of the digits of 1944 (and where $112 = 7 × 4 × 4$ has a reduced totient of 60). 432 is also the twenty-third interprime between twin primes, and the eighth such number whose adjacent prime numbers have prime indices that add to a prime number ($2012 = 2^{2} × 503$, the thirty-ninth prime); 348, the composite index of 432, is itself the sixth interprime whose adjacent prime numbers have indices that add to a prime number ($1176 = s(744)$, the thirty-fourth prime). Importantly, $1632 = 1176 + 456$ is an Achilles number, a powerful number (the thirty-fourth) that is itself not a perfect power, like 864 (the eleventh index, that is twice its value), and 288 (the totient of 864), as well as 108 (a fourth of 432), and 1944. (432 is the sixth member, where the sixth triangular number is 21, itself the index of 1944 in the list of Achilles numbers. Also, $576 = 24^{2}$, where $613 + 744 = 1357$, or in other words $744 − 613 = 131$.) On the other hand, 181, the index of 744 as an abundant number, is the sixteenth full repetend prime in base ten, and the composite index of 232, that is the sum of the first five interprimes to lie in-between twin primes with prime sums from respective indices (i.e., 138, 72, 12, 6, 4), as with 432 and 348. 109 is the tenth full repetend prime, repeating 108 digits whose sum of repeating digits is 486, a fourth of 1944, where 487 is the thirty-third full repetend prime (and 811 the forty-ninth, one less than the sum of the 180 repeating digits of 181; with $585 = 8^{0} + 8^{1} + 8^{2} + 8^{3}$).

Convolution of Fibonacci numbers
744 is the twelfth self-convolution of Fibonacci numbers, which is equivalently the number of elements in all subsets of $$\{1,2,3,...,11\}$$ with no consecutive integers.

In graph theory
The number of Euler tours (or Eulerian cycles) of the complete, undirected graph $$K_{6}$$ on six vertices and fifteen edges is 744. 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). 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.

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, whose average of divisors is 744 (as with thrice any prime number $$3p$$, the average of divisors will be $$p + 1$$). This value is a difference of 1110 from 3339, which is the sum of seven hundred and forty-two (742) repeating digits of the reciprocal of 743, as the forty-eighth full repetend prime in decimal (with the smallest number to have a Euler totient of 744 being 1119).

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), that also represents the number of irreducible trees with fifteen vertices. While for the complete undirected graph $$K_{5}$$ there are 264 directed Eulerian circuits, 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 $$T(n,k)$$ of length $$k$$ on $$n$$ labeled vertices.

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 $$\{\{1,4\},\{2,5\},\{3,6\}\}$$ on vertices labelled $$1$$ through $$6$$ in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.

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 $$M$$ 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; as well as the number of cliques in the 7-triangular graph, where every subset of two distinct vertices in a clique are adjacent. 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.

In particular, 456 is the aliquot sum of 264, the only number to have this value for $$s(n).$$

j-invariant
The j–invariant holds as a Fourier series $1⁄p$–expansion,

$$j(\tau) = q^{-1} + 744 + 196\,884 q + 21\,493\,760 q^2 + 864\,299\,970 q^3 + \cdots$$

where $$q = e^{2\pi i\tau}$$ and $$\tau$$ the half-period ratio of an elliptic function.

The $p − 1$–invariant can be computed using Eisenstein series $$E_4(\tau)$$ and $$E_6(\tau)$$, such that: $$j(\tau) = 1728 \frac{E_4(\tau)^3}{E_4(\tau)^3 - E_6(\tau)^2} ,$$ where specifically, these Eisenstein series are equal to $$E_4(\tau) = 1 + 240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n}$$ and $$E_6(\tau) = 1- 504\sum_{n=1}^\infty \frac{n^5 q^n}{1-q^n}$$ with $$q = \text{exp}(2\pi i \tau)$$.

The respective $147 = 48 + 49 + 50$–expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −504, respectively; where specifically the sum and difference between the absolute values of these two numbers is $81 = 9^{2}$ and $744 − 383 = 361 = 19^{2}$. Furthermore, when considering the only smaller even (here, non-modular) series $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty \frac {nq^n}{1-q^n},$$ the sum between the absolute value of its constant multiplicative term (24) and that of $$E_2(\tau)$$ (240) is equal to 264 as well. The 16th coefficient in the expansion of $$E_2(\tau)$$ is −744, as is its 25th coefficient.

Alternatively, the j–invariant can be computed using a sextic polynomial as: $$j(\tau) = 256 \cdot \frac {\bigl(1-\lambda(1-\lambda)\bigr)^3}{\bigl(\lambda(1-\lambda)\bigr)^2} = 256 \cdot \frac{\left(1-x\right)^3}{x^2}$$ where $$\lambda$$ represents the modular function, with $76 = 2^{2} × 19$.

Almost integers
Also, the almost integer

$$e^{\pi \sqrt{163}} \approx 262\,537\,412\,640\,768\,743.999\,999\,999\,999\,250\,072\,59 \approx 640\,320^3 + 744.$$

This number is known as Ramanujan's constant, which is transcendental. Mark Ronan and other prominent mathematicians have noted that the appearance of $$163$$ 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 $$d$$ such that the imaginary quadratic field $\Q\left[\sqrt{-d}\right]$ has class number of $$1$$ (equivalently, the ring of integers over the same algebraic number field have unique factorization). $$\mathrm {F_{1}}$$ 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. The linear term of error $$O$$ for Ramanujan's constant is approximately,

$$\frac{-196\,884}{e^{\pi \sqrt{163}}} \approx \frac{-196\,884}{640\,320^3+744} \approx -0.000\,000\,000\,000\,75 ,$$

where $$196\,884$$ is the value of the minimal faithful complex dimensional representation of the Friendly Giant $$\mathrm {F_{1}}$$, the largest sporadic group. $$\mathrm {F_{1}}$$ contains an infinitely graded faithful dimensional representation equivalent to the $$q$$ coefficients of the series of the q-expansion of the j-invariant. Specifically, all three common prime factors $$(2,3,5)$$ that divide the Euler totient, sum-of-divisors, and reduced totient of $$744$$ are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups $$(\mathrm {F_{1}},\text{ }\mathrm {B},\text{ }\mathrm {F_{3}},\text{ }\mathrm {Ly},\text{ }\mathrm {ON},\text{ }\mathrm {J_{4}})$$ whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, $$31$$; three of these belong inside the small family of six pariah groups that are not subquotients of $$\mathrm {F_{1}}.$$ The largest supersingular prime that divides the order of $$\mathrm {F_{1}}$$ is $$71,$$  which is the eighth self-convolution of Fibonacci numbers, where $$744$$ is the twelfth.

Other almost integers
The largest three Heegner numbers with $$d > 19$$ also give rise to almost integers of the form $$e^{\pi \sqrt{d}}$$ which involve $$744$$. In increasing orders of approximation,

$$\begin{align} e^{\pi \sqrt{19}} &\approx {\color{white}000\,0}96^3+744-0.22\\ e^{\pi \sqrt{43}} &\approx {\color{white}000\,}960^3+744-0.000\,22\\ e^{\pi \sqrt{67}} &\approx {\color{white}00}5\,280^3+744-0.000\,0013\\ e^{\pi \sqrt{163}} &\approx 640\,320^3+744-0.000\,000\,000\,000\,75 \end{align} $$

Square-free positive integers over the negated imaginary quadratic field with class number of $$2$$ also produce almost integers for values of $$d$$, where for instance there is $$199\,148\,648^2 - 0.000\,97\ldots \approx e^{\pi \sqrt{148}} + (8 \times 10^{3} + 744).$$

$11 × 31$
In the list of thirteen integers $(37 + 41 + 43 + 47 + 53 + 59 + 61)$ that yield almost integers — with nearness $4^{4} + 4^{3} + 4^{2} + 4^{1} + 4^{0}$ for values $s(341) = 43$ — the smallest $7 × 31$ of these is 25, that is also the smallest composite totative of 744. Less than 103, the largest such $3 mod 4$ is $5 × 31$, where 719 represents the 128th indexed prime number; the sum generated by these two $49 = 7^{2}$ is in equivalence with $(G, X)$ The middle indexed value in between these two bounds is 148, the twelfth square-free positive integer $81$ over the negated imaginary quadratic field of class number 2, followed by 163 and 232, the latter of which is the fourteenth square-free positive integer $X$ over the imaginary quadratic field $G$ of class number 2 (only these two numbers 148 and 232 for $n^{2} + n + 41$ in this field of class number 2 yield almost integers with $p(k), ..., p(k+r)$, where 163 is the largest over the same field with class number 1). In this same list, the three distinct non-supersingular primes (37, 43, 67) that only divide orders of pariah groups are also $r ≥ 1$ (less than 148) that yield almost integers of the form $±1 mod 6$. In the almost integer representation for $713 − 589 = 124$ specified, $713 − 527 = 186$, the class $713 − 217 = 496$ j−invariant of order $713 − 155 − 62$, that is also a root in $746 = 2 × 373 = 1^{5} + 2^{4} + 3^{6} = 2! + 4! + 6!$, a polynomial for supersingular 2-isogeny graphs with loops (it is also a root in $1119 − 1492$). The remaining two $1119 + 1492 + 2238 = 4849$ (58, 74) in the list of these almost integers with nearness $6 × 28$ of $5! × (2!)^{3}$ less than 148 generate a sum of $149 + 151 + 157 + 163 + 167 + 173$ equal to the prime index of 743, the largest prime totative of 744 (and, where $120 + 136 = 256 = 2^{8}$).

$29 = 1 + 2 + 3 + 5 + 7 + 11 = 2^{2} + 3^{2} + 4^{2}$
For almost integers with nearness $n = 8$, the smallest such number is $k = 6$ with ${1, 2, ..., 31}$ of 6, approximately equal to 2197.99087. Whereas 2198 holds eight divisors that produce an arithmetic mean of 474 (and where 474 holds a sum-of-divisors equal to 960, also the Zumkeller half from the set of divisors of 744), 2199 is the sixteenth perfect totient number, with an aliquot sum of 737 equivalent with the largest composite totative of 744. Also, $r_{16}$. There are a total of twenty-six almost integers with $(3307, 3313, 3319)$ where $p$ and $p − 6$, where the largest $p + 6$ holds a value of 986. The sum between the upper and lower bounds $2^{p - 1}(2^{p} − 1)$ less than one thousand with this almost integer degree of nearness is equal to $p = 5$. Regarding 992, the numbers 336 (the totient of 1176, which is the sum-of-divisors of 744), 496 (the 31st triangular number and third perfect number), and 525 — the sum of all prime numbers that divide the orders of the twenty-six (or twenty-seven) sporadic groups, equal to the sum of the dimensions of all five exceptional Lie algebras — are the first three of only four composites to have a sum-of-divisors of 992. Furthermore, the fifth such number is 775, whose composite index is 637, equal to the sum of all prime factors (inclusive of multiplicities) in the order of the largest sporadic group, $256 = 2^{8}$ $1000 = 10^{3}$. In Chowla's function that counts the sum-of-divisors of $488 = 240 + 248.$ except for 1 and $14 = 7 + 7$, 525 holds a value of 466 that is the prime index of 3313, equal to the sum of all seven numbers (240, 350, 366, 368, 575, 671, and 743) to hold a sum-of-divisors of 744; with 240 the totient of 525 as well (the 13th of 31 numbers to hold this value).

Square-free integers of class number 3
Regarding square-free integers of class number $49 = 7 × 7$, there are a total of sixteen (or twenty-five when including non-maximal orders), the largest with value of 907 that is the 155th indexed prime number, equal to the sum $109 = 42 + 28 + 18 + 10 + 5 + 3 + 2 + 1 + 0$, that is also the greater of consecutive primes (887, 907) that generate the seventh largest record prime gap (of 20). The previous greatest prime gap is of 18 set by (523, 541) (following the thirtieth and thirty-first prime numbers 113 and 127 with a gap of 14), the 99th and 100th prime numbers, the former also the composie index of 132, and the latter the tenth star number and 53rd number to return $5 + 7 + ... + 29$ for the Mertens function (where 427 is the 50th, and 163 the 13th).

In the full list of largest square-free positive integers with class numbers $1455 = 1176 + 279$, ten of these maximum values are held by larger integers when non-maximal orders in their respective classes are included, where nine of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is $6$ of class number $25 = 5^{2}$. In the difference between the largest class $n$ and $σ(744) = 1920$ square-free integers, there is $n$ Of the square-free integers of class number three, only 59 produces an almost integer of the form $3^{n}$ with $3^{3}$ (for integers of class number $3^{4}$, the largest such number is 1555 — whose square divides $279000 = 279 × 1000$ — while the ninth such largest discriminant is 155).

Riemann zeta function
The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822...}. The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the $1$ member in this sequence. Taking the sum of the three terms that lie between 71 and 744 (i.e. 130, 235, 420) yields $9653449 = 3107^{2} = (13 × 239)^{2}$, whose aliquot sum is 163, the thirty-eighth prime number. 785 is the 60th number to return $1=1^{2}$ for the Mertens function, which also includes 163, the 13th such number. 785 is also the number of irreducible planted trees (of root vertex having degree one) with six leaves of two colors.

1106, the first member in the sum of consecutive composite numbers (in equivalence with $169 = 13^{2}$) that are divisible by the twenty-fifth prime number 97, is the smallest number to return a value of 14 in the inverse Mertens function, where 1106 lies between the 81st pair of sexy primes $y^{2} + 1 = 2x^{4}$. 14 is the floor (and nearest integer) of the imaginary part of the first non-trivial zero in the Riemann zeta function $(x, y) = (1, 1)$, and the fourteenth indexed floor value is 60 — sixty is the smallest number with exactly twelve divisors, and there are only two numbers which have a sum-of-divisors of 60: 24 and 38, whose sum is 62 (which is the tenth largest divisor of 744); also, three of nine numbers with totients of 60 are also divisors of 744 (93, 124, and 186, that add to 403). On the other hand, the difference $(13, 239)$, where the sixtieth floor value in the imaginary part of its nontrivial zero in $239⁄(13^{2})$ is the largest Heegner number 163.

In relation to Robin's theorem
The twentieth prime number is 71, where 31 is the eleventh; in turn, 20 is the eleventh composite number that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163. 71 is also part of the largest pair of Brown numbers $√2$, of only three such pairs; where in its case $16 arctan (1⁄5) − 4 arctan (1⁄239)$. Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744; where $5 × 239 = 1195$ holds an aliquot sum of 611, which is the composite index of 744.

5040 is the nineteenth superabundant number that is also the largest factorial that is a highly composite number, and the largest of twenty-seven numbers $52 = 13 × 4 = 26 × 2$ for which the inequality $1:1:3$ holds, where $3:1:1$ is the Euler–Mascheroni constant; this inequality is shown to fail for all larger numbers if and only if the Riemann hypothesis is true (known as Robin's theorem). 5040 generates a sum-of-divisors $1:3:1$ that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers $279 = 3^{2} × 31$ in Robin's theorem to hold $16 = 4^{2}$ such that $62 = 2 × 31$ | $63 + 44 = 107$ for any subset of divisors $181 = 90 + 91$ of $81 = 9^{2}$; the only other such number is 240: Where also ${1, 4, 9, 19, 37, 73, 143}$, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is $n$, which is the middle indexed composite number congruent $11 + 13 + 17 + 19 + 23 + 29 + 31$ less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number $(11 × 13)$ after 744 such that $3^{4} + 4^{4} + 5^{4} + 6^{4} = 7^{4} − 143$ is $3^{2} + 4^{2} = 5^{2}$. Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these $3^{3} + 4^{3} + 5^{3} = 6^{3}$ is in equivalence with $3^{1} = 4^{1} − 1$ The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence 2520, where $0 = 3^{0} − 1$ represents a Zumkeller half from the set of divisors of 744, with $27 = 3^{3}$ (and while $983 + 17 = 1000$, where 720 is the smallest number with thirty divisors, equal to $2 × 279 + 432 = 7 + 983$, a difference between the aliquot sum of 744, and the only number to have an aliquot sum of 744; wherein 720 is also in equivalence with $18 × 18$). $1⁄19$ is divisible by the first twelve non-zero integers, except for 11.

D4 and F4
$$744$$ is theta series coefficient $$25$$ of four-dimensional cubic lattice $$\mathbb {D}_{4} \cong \mathbb {(Z^{4})^{+}}.$$  On the other hand, in the theta series of the four-dimensional body-centered cubic lattice $$\mathbb {F}_{4}$$ — whose geometry with $$\mathbb {D}_{4}$$ defines Hurwitz quaternions of even and odd square norm as realized in the $$24$$–cell honeycomb that is dual to the $$16$$–cell honeycomb (and, as a union of two self-dual tesseractic $$8$$–cell honeycombs) — the sixteenth $$144 = 12^{2}$$ coefficient is the seven hundred and forty-fourth coefficient in the series; with coefficient index $$144$$ the forty-ninth non-zero norm.

$$\mathbb{D}$$$1⁄2$ ≅ $$\mathbb{Z}^4$$ as a lattice is the union of two $$\mathbb{D}_4$$ lattices, where the associated theta series of $$\mathbb{Z}^4$$ has 744 as its 50th indexed coefficient (as with theta series of $$\mathbb{D}_4$$), and where its twenty-fifth coefficient is 248, which is the most important divisor of 744. Also, in this theta series of $$\mathbb{Z}^4$$, the preceding 49th coefficient is 456, the only number to hold an aliquot sum of 744, where $62 = 31 × 2$, 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 $$\mathbb{D}_4$$, that is realized in the 16-cell honeycomb, all $3 + 61 = 64 = 8^{2}$ indexed coefficients (i.e. 25, 50, 100, 200, 400, ...) are 744. For both the theta series of $$\mathbb{D}_4$$ and $$\mathbb{Z}^4$$, the 456th coefficient is 1920, the sum-of-divisors of 744.

For the theta series of the four-dimensional $$\mathbb{F}_4$$ 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; 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 — the smallest of these is the twenty-sixth prime number, while the largest $13^{2} + 13 + 1 = 183$ has a sum of prime factors that is 17, the seventh prime number; and where $x^{2} + xy + y^{2}$ and $(a,b)$, with $(3, 40), (8, 37), (15, 32)$, the 447th indexed composite number. More deeply, for the theta series of $$\mathbb{F}_4$$ 744 is a prime-indexed coefficient over its first six indices less than $(23, 25)$ (respectively, the 458th, 526th, 742nd, 799th, 842nd, and 1141st prime indices, with a sum of 4058; the latter in equivalence with $2232 = 744 × 3$). The first composite coeff. index in the series with coefficient 744 is its seventh index $1632 = 1176 + 456$, 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, are also prime-indexed coefficients over their first two coefficient indices (346th, 364th, and 1098th, 1159th, respectively).

In four-dimensional space, three-dimensional cell facets of the three-largest of six regular $$4$$–polytopes (the octaplex, dodecaplex, and tetraplex) collectively number $$24 + 120 + 600 = 744.$$

E8 and the Leech lattice
Within finite simple groups of Lie type, exceptional Lie algebra $$\mathfrak{e_{8}}$$ holds a minimal faithful representation in two hundred and forty-eight dimensions, where $$248$$ divides $$744$$ thrice over. 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 $$\mathfrak{e}_{8}$$ as well as those of $$\mathfrak{e}_{7}$$ and $$\mathfrak{e}_{6}$$ respectively coincide with the three largest conjugacy classes of $$\mathrm {F_{1}}$$; where also the corresponding McKay–Thompson series $$j(3\tau)^{1/3}$$ of sporadic Thompson group $$\mathrm {Th}$$ holds coefficients representative of its faithful dimensional representation (also minimal at $$\operatorname {dim} 248$$) whose values themselves embed irreducible representation of $$\mathfrak{e_{8}}$$. In turn, exceptional Lie algebra $$\mathfrak{e_{8}}$$ is shown to have a graded dimension $$j(q)^{1/3}$$ whose character $$\chi$$ lends to a direct sum equivalent to,


 * $$\chi_{e_{8} \oplus e_{8} \oplus e_{8}} (q) = J(q) + 744 = j(q),$$ where the CFT probabilistic partition function for $$\mathrm {F_{1}}$$ is $$J(q)$$ of character $$\chi_{F_{1}}.$$

The twenty-four dimensional Leech lattice $$\Lambda_{24}$$ in turn can be constructed using three copies of the associated $$\mathbb {E_{8}}$$ lattice and with the eight-dimensional octonions $$\mathbb {O}$$ (see also, Freudenthal magic square), where the automorphism group of $$\mathbb {O}$$ is the smallest exceptional Lie algebra $$\mathfrak{g_{2}}$$, which embeds inside $$\mathfrak{e_{8}}$$. In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from $$V_1$$ (as $$\mathbb {C}_{24}$$) with a central charge $$c$$ of $$24$$, out of a total seventy-one such modular invariant conformal field theories of holomorphic VOAs of weight one. Known as Schellekens' list, these algebras form deep holes in $$V_{\Lambda_{24}}$$ whose corresponding orbifold constructions are isomorphic to the moonshine module $$V_{2}$$♮ that contains $$\mathrm {F_{1}}$$ as its automorphism; of these, the second and third largest contain affine structures $$E^3_{8,1}$$ and $$D_{16,1}E_{8,1}$$ that are realized in $$\operatorname {dim}744$$.

Of these, $$V_{E_{8}^{3}}$$ is isomorphic to the tensor product $$V_{E_{8}} \otimes V_{E_{8}} \otimes V_{E_{8}}$$; also, affine structure $$D_{16}$$ is different from $$D_{16}^{+}$$, which is associated with even positive definite unimodular lattice $$\mathbb{D}_{16}^{+}$$. $$E_{8,1}^{3}$$ and $$D_{16,1}E_{8,1}$$ are associated with codes $$e_{8}^{3}$$ and $$d_{16}e_{8}$$ that are two of only nine in-equivalent doubly even self-dual codes of length 24 and weight 4. The largest of these vertex operator algebras ($$V_{D_{24}}$$) is realized in dimension $$1128$$, where successively halving its dimensional space leads to a $$70 \tfrac{1}{2}$$ dimensional space.

Selfie number
It is a "selfie number", where $$744 = (7 - 4)!! + 4!$$, such that it can be expressed using just its digits (which are only used once, and from left to right) alongside the operators $2475 = 4419 − 1944$ (with concatenation allowed).

This, in likeness of $$144 = (1 + 4)! + 4!$$, that is the Euler totient of 456. Where the totient of 744 is 240, that of 456 is 144. 187 is the composite index of 240, where 187 is the 144th composite number. In turn, the sum-of-divisors of 187 is $167 = 83 + 84$, which is the 168th composite number. Also, the reduced totient of 456 is 36.

Pentagonal numbers
$$744$$ is also the sum of consecutive pentagonal numbers, $$P_{11} + P_{12} + P_{13} = 210 + 247 + 287. $$

Where the smallest non-unitary pentagonal pyramidal number is 6, the eleventh is 726 = 6! + 6, and the twenty-fourth is 7200, which is a number with a Euler totient value of 1920, and a reduced totient of 120.

Magic figures
$$744$$ is the magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between $$41$$ and $$223$$ inclusive.

The magic square is:

$$\begin{bmatrix} 139 & 113 & 151 & 131 & 83 & 127 \\ 223 & 149 & 89 & 47 & 157 & 79 \\ 173 & 103 & 181 & 167 & 59 & 61 \\ 67 & 137 & 53 & 97 & 211 & 179 \\ 101 & 199 & 73 & 109 & 71 & 191 \\ 41 & 43 & 197 & 193 & 163 & 107 \end{bmatrix}$$

This is the second-smallest magic constant for a $139 = 69 + 70$ magic square consisting of thirty-six consecutive prime numbers, where the sum between the smallest and largest prime numbers in this square is equal to $432 = 3^{3} + 4^{3} + 5^{3} + 6^{3}$. The smallest such constant is $1944 = 1200 + 744$ whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal; specifically, 22 and 264 are respectively the twelfth and fourteenth numbers $1200 = 456 + 744$ whose squares are undulating in decimal, while the thirteenth and penultimate such known number is 26. The smallest possible magic constant of an $σ_{}(456)$ magic square consisting only of distinct prime numbers is 120, from an $288 = 180 + 108$ of $1$, a value equal to the arithmetic mean of all sixteen divisors of 744; otherwise, for $n$, the smallest magic constant for a six-by-six square with distinct prime numbers is 432, also the abundance of 744.

An $6n + 1 = 49$ magic square that is normal has a magic constant of 671, which is the sixth and largest composite number to have a sum-of-divisors equal to 744.

Polygonal regions
744 is the number of non-congruent polygonal regions in a regular $$36$$–gon with all diagonals drawn. Otherwise, $6$ is the least possible number of diagonals of a simple convex polyhedron with thirty-six faces; for sixteen and twenty faces, there are respectively a minimum of 132 and 240 diagonals (values which represent the prime index of the largest prime totative of 744 and the count of all its totatives), where $0 ≤ n < 8$, or half 744 (alongside respective indices that generate a sum of 36).

Perfect rectangles
There are seven hundred and forty-four ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle. Meanwhile, 456 (the only number to have an aliquot sum of 744) is the maximum number of unit squares — only joined at corners — that can be inscribed inside a $$ 40 \times 40$$ square (of area $n = 6$$n = 5$).