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

It can be represented as the sum of nonconsecutive factorials $k!$ ,^[1] as the sum of four consecutive primes $p$ ,^[2] and as the product of sums of divisors $\sigma (n)$ of consecutive integers $n$ ;^[3] respectively:

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

744 is equal to the sum of a subset of its divisors (1 + 2 + 4 + 24 + 62 + 93 + 124 + 186 + 248), as a semiperfect number.^[4] It is also an abundant number, because the sum of its proper divisors is greater than itself.^[5]^[6]

The number partitions of the square of seven (49) into prime parts is 744,^[7] as is the number of partitions of 48 into at most four distinct parts.^[8] The radical $186 = 2 \times 3 \times 31$ of 744^[9] has an arithmetic mean of divisors equal to 48,^[10] where the sum between the three distinct prime factors of 744 is $62 = 36$ . 744 is also a practical number,^[11] and the first number to be the sum of nine cubes in eight or more ways,^[12] as well as the number of six-digit perfect powers in decimal.^[13]

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.

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.^[14] 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^[15] (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),^[16] 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 $\lambda (744)=30=2\times 3\times 5$ .^[17]

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.^[18] 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).^[19] On the other hand, only three numbers hold a totient of 744; they are 1119, 1492, and 2238.^[20]

φ(σ(744)) = 744

744 is the sixth number $n$ whose totient value has a sum-of-divisors equal to $n$ .^[21] Otherwise, the aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is 1176^[22] which is the forty-eighth triangular number,^[23] and the binomial coefficient $\operatorname {C(49,2)}$ present inside the forty-ninth row of Pascal's triangle.^[24]

In total, only seven numbers have sums of divisors equal to 744; they are: 240, 350, 366, 368, 575, 671, and 743.^[25]

Only one number has an aliquot sum that is 744, it is 456.^[22]

Zumkeller number

744 is also a Zumkeller number whose divisors can be partitioned into two disjoint sets with equal sum: 960.^[26] 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)

Properties specific to particular bases

In binary, 744 is a pernicious number, as its digit representation (1011101000₂) contains a prime count (5) of ones.^[27]

Meanwhile, in septenary 744 is palindromic: 744₁₀ = 2112₇.^[28]

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.^[29]^[30]^[31]

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.^[32]

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.^[33] 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.^[34]

Almost integers

Ramanujan's constant is the transcendental almost integer^[35]^[36]

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

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 $d$ :^[37]^{: p.20–23}

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

D₄ lattice

$744$ is theta series coefficient $25$ of the four-dimensional cubic lattice $\mathbb {D} _{4}\cong \mathbb {(Z^{4})^{+}}$ , or equivalently, the number of Hurwitz integer quaternions with norm 25.^[38]^[39]

E₈ and the Leech lattice

The exceptional Lie algebra ${\mathfrak {e_{8}}}$ has a graded dimension $j(q)^{1/3}$ ^[40] whose character $\chi$ lends to a direct sum equivalent to,^[41]^{: p.7, 9–11}

\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}}.

^[34]

The twenty-four dimensional Leech lattice $\Lambda _{24}$ can be constructed using three copies of the associated $\mathbb {E_{8}}$ lattice^[42]^[43]^{: pp.233–235} and with the eight-dimensional octonions $\mathbb {O}$ (see also, Freudenthal magic square),^[44] 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.^[45] 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;^[46] 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 $\operatorname {dim} 744$ .

In the Riemann zeta function

Relation to Robin's theorem

The twentieth prime number is 71, where 31 is the eleventh; in turn, 20 is the eleventh composite number^[47] 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 $72 - 1 = 5040$ .^[48]^[49] Consequently, both 5040 and 5041 can be represented as sums of non-consecutive factorials, following 746, 745, and 744;^[1] where $5040 + 5041 = 10081$ holds an aliquot sum of 611, which is the composite index of 744.^[47]

5040 is the nineteenth superabundant number^[50] that is also the largest factorial that is a highly composite number,^[51] and the largest of twenty-seven numbers $n$ for which the inequality $σ(n) \geq e γ n loglog n$ 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).^[52] 5040 generates a sum-of-divisors $19344 = 13 \times 31 \times 48$ 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 $744 m$ | $σ(n)$ for any subset of divisors $m$ of $n$ ; the only other such number is 240:^[53]

$19344 \div 26 = 744$
$9672 \div 13 = 744$
$1488 \div 2 = 744$

Where also $19344 \div 78 = 248$ , with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is $403 = 13 \times 31$ , which is the middle indexed composite number congruent $\pm1 mod 6$ 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 $n$ .^[21] 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,^[26] with $σ(720) = 2418$ ^[22] (and while $720 + 24 = 744 = 6! + 24$ , where 720 is the smallest number with thirty divisors,^[54] 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 $σ(264) = 456 + 264$ ).^[22]^[25] $5040 = 7! = 10 \times 9 \times 8 \times 7$ is divisible by the first twelve non-zero integers, except for 11.

Other properties

Selfie number

It is a "selfie number", where $744=(7-4)!!+4!$ ,^[55]^[56] such that it can be expressed using just its digits (which are only used once, and from left to right) alongside the operators $+, -, \times, \div, a b, \sqrt, !$ (with concatenation allowed).

This, in likeness of $144=(1+4)!+4!$ , that is the Euler totient of 456.^[20] Where the totient of 744 is 240, that of 456 is 144.^[20] 187 is the composite index of 240, where 187 is the 144th composite number.^[47] In turn, the sum-of-divisors of 187 is $216 = 6 3$ ,^[25] which is the 168th composite number. Also, the reduced totient of 456 is 36.^[17]

Pentagonal numbers

$744$ is also the sum of consecutive pentagonal numbers,^[57]^[58]

$P_{11}+P_{12}+P_{13}=210+247+287.$

Where the smallest non-unitary pentagonal pyramidal number is 6, the eleventh is 726 = 6! + 6, and the twenty-fourth is 7200,^[59] which is a number with a Euler totient value of 1920,^[20] and a reduced totient of 120.^[17]

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.^[60]

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 $6 \times 6$ 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 $484 = 22 2$ ^[61] whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal;^[22] 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.^[62]^[63]^[64]^{: pp.159, 160} The smallest possible magic constant of an $n \times n$ magic square consisting only of distinct prime numbers is 120, from an $n$ of $4$ ,^[65]^[66] a value equal to the arithmetic mean of all sixteen divisors of 744;^[67]^[10] otherwise, for $n = 6$ , the smallest magic constant for a six-by-six square with distinct prime numbers is 432,^[65] also the abundance of 744.^[6]

An $11 \times 11$ magic square that is normal has a magic constant of 671,^[68] which is the sixth and largest composite number to have a sum-of-divisors equal to 744.^[25]

Polygonal regions

744 is the number of non-congruent polygonal regions in a regular $36$ –gon with all diagonals drawn.^[69]

Perfect rectangles

There are 744 ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle.^[70]

References

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^ Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011). "Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis". Integers. 11 (6). Berlin: De Gruyter: 755 (A33). arXiv:1110.5078. doi:10.1515/INTEG.2011.057. MR 2798609. S2CID 14573608. Zbl 1235.11082.
^ Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.
^ Taneja, Inder J. (2016). "Selfie Numbers - IV: Addition, Subtraction and Factorial" (PDF). RGMIA Research Report Collection. 19: 1−4 (Article 163).
^ Sloane, N. J. A. (ed.). "Sequence A193069 (Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.
^ Sloane, N. J. A. (ed.). "Sequence A129863 (Sums of three consecutive pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers: a(n) equal to n^2*(n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-09.
^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
^ Sloane, N. J. A. (ed.). "Sequence A122875 (Numbers whose square is undulating.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
^ Sloane, N. J. A. (ed.). "Sequence A016073 (Undulating squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.
^ Pickover, Clifford A. (1995). "Chapter 20: The Undulation of the Monks". Keys to Infinity. New York, NY: John Wiley & Sons, Inc. pp. xviii, 1−332. ISBN 978-0471193340. OCLC 31710129. S2CID 121761396.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A164843 (The smallest magic constant of an n X n magic square with distinct prime entries.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-05.
^ Makarova, Natalia V. "Least Magic Squares of Prime Numbers (Part I)". klassikpoez.narod.ru (in Russian). Archived translation in English. Retrieved 2023-12-07.
^ Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.
^ Sloane, N. J. A. (ed.). "Sequence A187781 (Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.
^ Sloane, N. J. A. (ed.). "Sequence A002839 (Number of simple perfect squared rectangles of order n up to symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[N-CF-1] Sloane, N. J. A. (ed.). "Sequence A060112 (Sums of nonconsecutive factorial numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[2] Sloane, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[3] Sloane, N. J. A. (ed.). "Sequence A083539 (a(n) is sigma(n) * sigma(n+1) as the product of sigma-values for consecutive integers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[4] Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.

[5] Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.

[AbunoN-6] Sloane, N. J. A. (ed.). "Sequence A033880 (Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-29.

[PartNPP-7] Sloane, N. J. A. (ed.). "Sequence A000607 (Number of partitions of n into prime parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[8] Sloane, N. J. A. (ed.). "Sequence A347586 (Number of partitions of n into at most 4 distinct parts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[RadN-9] Sloane, N. J. A. (ed.). "Sequence A007947 (Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-31.

[ArithNum-10] Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[PractNum-11] Sloane, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-03.

[12] Sloane, N. J. A. (ed.). "Sequence A345547 (Numbers that are the sum of nine cubes in eight or more ways.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[13] Sloane, N. J. A. (ed.). "Sequence A075308 (Number of n-digit perfect powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[RecSigm-14] Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[15] Sloane, N. J. A. (ed.). "Sequence A006002 (a(n) equal to n*(n+1)^2/2 (Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ...))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[16] Sloane, N. J. A. (ed.). "Sequence A081377 (Numbers n such that the set of prime divisors of phi(n) is equal to the set of prime divisors of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[redTot-17] Sloane, N. J. A. (ed.). "Sequence A002322 (Reduced totient function psi(n): least k such that x^k is congruent 1 (mod n) for all x prime to n; also known as the Carmichael lambda function (exponent of unit group mod n); also called the universal exponent of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-27.

[18] Sloane, N. J. A. (ed.). "Sequence A038509 (Composite numbers congruent to +-1 mod 6.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[Osiris-19] Sloane, N. J. A. (ed.). "Sequence A319274 (Osiris or Digit re-assembly numbers: numbers that are equal to the sum of permutations of subsamples of their own digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[Etot-20] Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[SigmPhi-21] Sloane, N. J. A. (ed.). "Sequence A018784 (Numbers n such that sigma(phi(n)) is n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[AliSum-22] Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[TriangNum-23] Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular number: a(n) is the binomial(n+1,2) equivalent to n*(n+1)/2 that is 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[24] Sloane, N. J. A. (ed.). "Sequence A017765 (Binomial coefficients C(49,n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-02.

[SigmaSoD-25] Sloane, N. J. A. (ed.). "Sequence A000203 (a(n) equal to sigma(n), the sum of the divisors of n. Also called sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[Zumk-26] Sloane, N. J. A. (ed.). "Sequence A083207 (Zumkeller or integer-perfect numbers: numbers n whose divisors can be partitioned into two disjoint sets with equal sum.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[27] Sloane, N. J. A. (ed.). "Sequence A052294 (Pernicious numbers: numbers with a prime number of 1's in their binary expansion.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-28.

[PalindB7-28] Sloane, N. J. A. (ed.). "Sequence A029954 (Palindromic in base 7.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.

[29] Moree, Pieter (2004). "Convoluted Convolved Fibonacci Numbers" (PDF). Journal of Integer Sequences. 7 (2). Waterloo, Ont., CA: University of Waterloo David R. Cheriton School of Computer Science: 13 (Article 04.2.2). arXiv:math.CO/0311205. Bibcode:2004JIntS...7...22M. MR 2084694. S2CID 14126332. Zbl 1069.11004.

[ConvFib-30] Sloane, N. J. A. (ed.). "Sequence A001629 (Self-convolution of Fibonacci numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[31] Belbachir, Hacène; Djellal, Toufik; Luque, Jean-Gabriel (2023). "On the self-convolution of generalized Fibonacci numbers". Quaestiones Mathematicae. 46 (5). Oxfordshire, UK: Taylor & Francis: 841–854. arXiv:1703.00323. doi:10.2989/16073606.2022.2043949. MR 4592901. S2CID 119150217. Zbl 07707543.{{cite journal}}: CS1 maint: Zbl (link)

[32] Sloane, N. J. A. (ed.). "Sequence A350028 (Number of Euler tours of the complete graph on n vertices (minus a matching if n is even).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[33] Berndt, Bruce C.; Chan, Heng Huat (1999). "Ramanujan and the modular j-invariant". Canadian Mathematical Bulletin. 42 (4): 427–440. doi:10.4153/CMB-1999-050-1. MR 1727340. S2CID 1816362.

[GebMatt-34] Gaberdiel, Matthias R. (2007). "Constraints on extremal self-dual CFTs". Journal of High Energy Physics. 2007 (11, 087). Springer: 10–11. arXiv:0707.4073. Bibcode:2007JHEP...11..087G. doi:10.1088/1126-6708/2007/11/087. MR 2362062. S2CID 16635058.

[35] Sloane, N. J. A. (ed.). "Sequence A060295 (Decimal expansion of exp(Pi*sqrt(163)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-18.

[36] Barrow, John D. (2002). "The Constants of Nature". The Fundamental Constants. London: Jonathan Cape. p. 72. doi:10.1142/9789812818201_0001. ISBN 0-224-06135-6. S2CID 125272999.

[KlaiJan-37] Klaise, Janis (2012). Orders in Quadratic Imaginary Fields of small Class Number (PDF) (MMath thesis). University of Warwick Centre for Complexity Science. pp. 1–24. S2CID 126035072.

[ThD4-38] Sloane, N. J. A. (ed.). "Sequence A004011 (Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.

[JiH-39] Chun, Ji Hoon (2019). "Sphere Packings [2013]" (PDF). In Tsfasman, Michael; et al. (eds.). Algebraic Geometry Codes: Advanced Chapters. Mathematical Surveys and Monographs. Vol. 238. Providence, RI: American Mathematical Society. pp. 229−278. doi:10.1090/surv/238. ISBN 978-1-4704-5263-6. MR 3966406. S2CID 182109921. Zbl 1422.14004.

[40] Sloane, N. J. A. (ed.). "Sequence A121732 (Dimensions of the irreducible representations of the simple Lie algebra of type E8 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-05.

[TGann-41] Gannon, Terry (2006). "Introduction: glimpses of the theory beneath Monstrous Moonshine" (PDF). Moonshine beyond the monster: The bridge connecting algebra, modular forms and physics. Cambridge Monographs on Mathematical Physics. Cambridge, MA: Cambridge University Press. pp. 1–15. ISBN 978-0-521-83531-2. MR 2257727. OCLC 1374925688. Zbl 1146.11026.
"In particular, $4124 = 3875 + 248 + 1$ and $34752 = 30380 + 3875 + 2 \cdot 248 + 1$ , where 248, 3875 and 30380 are all dimensions of irreducible representations of $\mathbb {C}$ ."^: 6

[42] Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN 2196-9701. ISBN 978-1-4757-2016-7. MR 1541550.

[HSMRP-43] Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co. pp. 1–321. ISBN 9780521201254. MR 0027148. OCLC 472190910.

[44] Wilson, Robert A. (2009). "Octonions and the Leech lattice". Journal of Algebra. 322 (6): 2186–2190. doi:10.1016/j.jalgebra.2009.03.021. MR 2542837.

[45] Schellekens, Adrian Norbert (1993). "Meromorphic c = 24 conformal field theories". Communications in Mathematical Physics. 153 (1). Berlin: Springer: 159–185. arXiv:hep-th/9205072. Bibcode:1993CMaPh.153..159S. doi:10.1007/BF02099044. MR 1213740. S2CID 250425623. Zbl 0782.17014.

[46] Möller, Sven; Scheithauer, Nils R. (2023). "Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra". Annals of Mathematics. 197 (1). Princeton University & the Institute for Advanced Study: 261–285. arXiv:1910.04947. Bibcode:2019arXiv191004947M. doi:10.4007/annals.2023.197.1.4. MR 4513145. S2CID 204401905. Zbl 1529.17040.

[CompNum-47] Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-27.

[48] Sloane, N. J. A. (ed.). "Sequence A216071 (Brocard's problem: positive integers m such that m^2 equal to n! + 1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[49] Sloane, N. J. A. (ed.). "Sequence A085692 (Brocard's problem: squares which can be written as n!+1 for some n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[50] Sloane, N. J. A. (ed.). "Sequence A004394 (Superabundant [or super-abundant] numbers: n such that sigma(n)/n greater than sigma(m)/m for all m less than n, sigma(n) being A000203(n), the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[51] Sloane, N. J. A. (ed.). "Sequence A002182 (Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-10.

[RobTh-52] Sloane, N. J. A. (ed.). "Sequence A067698 (Positive integers such that sigma(n) greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-09.

[RobThRieH-53] Caveney, Geoffrey; Nicolas, Jean-Louis; Sondow, Jonathan (2011). "Robin's Theorem, Primes, and a New Elementary Reformulation of the Riemann Hypothesis". Integers. 11 (6). Berlin: De Gruyter: 755 (A33). arXiv:1110.5078. doi:10.1515/INTEG.2011.057. MR 2798609. S2CID 14573608. Zbl 1235.11082.

[LeastD-54] Sloane, N. J. A. (ed.). "Sequence A005179 (Smallest number with exactly n divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-06.

[55] Taneja, Inder J. (2016). "Selfie Numbers - IV: Addition, Subtraction and Factorial" (PDF). RGMIA Research Report Collection. 19: 1−4 (Article 163).

[56] Sloane, N. J. A. (ed.). "Sequence A193069 (Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-12.

[57] Sloane, N. J. A. (ed.). "Sequence A129863 (Sums of three consecutive pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[58] Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[59] Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers: a(n) equal to n^2*(n+1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-09.

[60] Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[PrimeMS-61] Sloane, N. J. A. (ed.). "Sequence A177434 (The magic constants of 6 X 6 magic squares composed of consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[62] Sloane, N. J. A. (ed.). "Sequence A122875 (Numbers whose square is undulating.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.

[63] Sloane, N. J. A. (ed.). "Sequence A016073 (Undulating squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-02.

[64] Pickover, Clifford A. (1995). "Chapter 20: The Undulation of the Monks". Keys to Infinity. New York, NY: John Wiley & Sons, Inc. pp. xviii, 1−332. ISBN 978-0471193340. OCLC 31710129. S2CID 121761396.

[MagPS-65] Sloane, N. J. A. (ed.). "Sequence A164843 (The smallest magic constant of an n X n magic square with distinct prime entries.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-05.

[66] Makarova, Natalia V. "Least Magic Squares of Prime Numbers (Part I)". klassikpoez.narod.ru (in Russian). Archived translation in English. Retrieved 2023-12-07.

[ArithNum0-67] Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

[68] Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) equal to n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-11.

[69] Sloane, N. J. A. (ed.). "Sequence A187781 (Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.

[70] Sloane, N. J. A. (ed.). "Sequence A002839 (Number of simple perfect squared rectangles of order n up to symmetry.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.

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744 (number)

Contents

Number theory

φ(n) and σ(n)

Totatives

φ(σ(744)) = 744

Zumkeller number

Properties specific to particular bases

Convolution of Fibonacci numbers

In graph theory

j-invariant

Almost integers

D₄ lattice

E₈ and the Leech lattice

In the Riemann zeta function

Relation to Robin's theorem

Other properties

Selfie number

Pentagonal numbers

Magic figures

Polygonal regions

Perfect rectangles

See also

References

744 (number)

Number theory

φ(n) and σ(n)

Totatives

φ(σ(744)) = 744

Zumkeller number

Properties specific to particular bases

Convolution of Fibonacci numbers

In graph theory

j-invariant

Almost integers

D4 lattice

E8 and the Leech lattice

In the Riemann zeta function

Relation to Robin's theorem

Other properties

Selfie number

Pentagonal numbers

Magic figures

Polygonal regions

Perfect rectangles

See also

References

D₄ lattice

E₈ and the Leech lattice