744 (number)

← 743 744 745 →
List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
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.

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:

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

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 = 5³, 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 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 6² = 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:

• (1, 2, 3, 4, 8, 12, 62, 124, 744)
• (6, 24, 31, 186, 93, 248, 372)

The index of 744 as a Zumkeller number (168) represents the product of the first two perfect numbers 6 × 28. 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 5! × (2!)³, equal to the sum of six consecutive prime numbers 149 + 151 + 157 + 163 + 167 + 173, between the 35th and 40th primes (it is the thirty-fifth such number). The fifteenth and sixteenth triangular numbers generate the sum 120 + 136 = 256 = 2⁸ that is the totient value of 960, and the number of partitions of 29 = 1 + 2 + 3 + 5 + 7 + 11 = 2² + 3² + 4² 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 n = 8 and k = 6. 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 (3307, 3313, 3319); it is respectively the 178th and 179th such prime p where p − 6 and p + 6 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 1:1:3 or 3:1:1 ratio with the positions of ones, which are in 1:3:1 ratio.
Its ones' complement is 100010111₂, equivalent to 279 = 3² × 31 in decimal, which represents the sum of GCDs of parts in all partitions of 16 = 4². It is also the number of partitions of 62 = 2 × 31 (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 63 + 44 = 107 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 181 = 90 + 91). 279, on the other hand, is the 58th number to lie between consecutive odd prime numbers (277, 281), with 81 = 9² 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 18 × 18 non-normal yet full prime reciprocal magic square based on its reciprocal (⁠1/19⁠).

Meanwhile, in septenary 744 is palindromic (2112₇),

Semiperfect number

744 is the 183rd semiperfect number, since it is equal to the sum of a subset of its divisors (e.g., 1 + 2 + 4 + 24 + 62 + 93 + 124 + 186 + 248).

Abundance and in full repetend primes

744 is the 181st abundant number, with an abundance of 432.

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.

In the 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 a(n) = 7 member in this sequence. Taking the sum of the three terms that lie between 71 and 744 (i.e. 130, 235, 420) yields 785 = 28² + 1, whose aliquot sum is 163, the thirty-eighth prime number. 785 is the 60th number to return 0 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 2229 = 743 × 3) 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 (1103, 1109). 14 is the floor (and nearest integer) of the imaginary part of the first non-trivial zero in the Riemann zeta function ζ, 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 38 − 24 = 14, where the sixtieth floor value in the imaginary part of its nontrivial zero in ζ is the largest Heegner number 163.

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 (71, 7), of only three such pairs; where in its case 7² − 1 = 5040. Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744; where 5040 + 5041 = 10081 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 n for which the inequality σ(n) ≥ e^γnloglogn holds, where γ 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 19344 = 13 × 31 × 48 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 n in Robin's theorem to hold σ(n) such that 744m | σ(n) for any subset of divisors m of n; the only other such number is 240:

• 19344 ÷ 26 = 744
• 9672 ÷ 13 = 744
• 1488 ÷ 2 = 744

Where also 19344 ÷ 78 = 248, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is 403 = 13 × 31, which is the middle indexed composite number congruent ±1 mod 6 less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number n after 744 such that σ(φ(n)) is n. 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 360 + 720 + 840 = 1920 is in equivalence with σ(744). The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence 2520, where 2520 − 840 − 720 = 960 represents a Zumkeller half from the set of divisors of 744, with σ(720) = 2418 (and while 720 + 24 = 744 = 6! + 24, where 720 is the smallest number with thirty divisors, equal to 1176 − 456, 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 σ(264) = 456 + 264). 5040 = 7! = 10 × 9 × 8 × 7 is divisible by the first twelve non-zero integers, except for 11.

j-invariant

The j–invariant holds as a Fourier series q–expansion,

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

where $q=e^{2\pi i\tau }$ and $\tau$ the half-period ratio of an elliptic function. The function without the constant term, $J(\tau )=j(\tau )-744$, is the partition function of the conformal field theory whose symmetries constitute the Monster group.

Almost integers

Ramanujan's constant is the transcendental 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 is an example of a more general phenomenon in which numbers of the form $e^{\pi {\sqrt {d}}}$ turn out to be nearly integers for special values of $d$:

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

D₄ and F₄

$744$ is theta series coefficient $25$ of four-dimensional cubic lattice $\mathbb {D} _{4}\cong \mathbb {(Z^{4})^{+}} .$

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

The exceptional Lie algebra ${\mathfrak {e_{8}}}$ has 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 $\Lambda _{24}$ 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 $E_{8,1}^{3}$ and $D_{16,1}E_{8,1}$ that are realized in $\operatorname {dim} 744$.

Other properties

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 +, -, ×, ÷, a^b, √, ! (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 216 = 6³, 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 6 × 6 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 41 + 223 = 264. The smallest such constant is 484 = 22² 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 n whose squares are undulating in decimal, while the thirteenth and penultimate such known number is 26. The smallest possible magic constant of an n × n magic square consisting only of distinct prime numbers is 120, from an n of 4, a value equal to the arithmetic mean of all sixteen divisors of 744; otherwise, for n = 6, the smallest magic constant for a six-by-six square with distinct prime numbers is 432, also the abundance of 744.

An 11 × 11 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.

Perfect rectangles

There are 744 ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle.

See also

• Moonshine theory – the study between the relationships held by F₁ and modular functions, in particular j(τ)

• Heegner numbers — 1, 2, 3, 7, 11, 19, 43, 67, and 163
• Supersingular primes — 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31, as well as 41, 47, 59, and 71
• The j-invariant — 1728 = 12³ = j(i) , one less than the first taxicab number 1729 (also known as the Ramanujan–Hardy number)