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Revision as of 19:45, 9 August 2024 editBeland (talk | contribs)Autopatrolled, Administrators237,084 edits →Other properties: consolidate short subsections, make formatting more consistent; retitle "Other relationships" because these are mostly not properties ''of'' 744 so much as it's a count in some other relationship← Previous edit Revision as of 19:56, 9 August 2024 edit undoAllan Nonymous (talk | contribs)Extended confirmed users, Pending changes reviewers4,538 edits Cite seems to have nothing to do with the text in question? Also, rewrote fact about the j-invartiant. I remember a quote about 744 in relation to this somewhere, I wonder if we can include it here.Tag: Visual editNext edit →
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== ''j''-invariant == == ''j''-invariant ==


The ] can be written as a ] ].<ref>{{Cite journal |last1=Berndt |first1=Bruce C. |author1-link=Bruce C. Berndt |last2=Chan |first2=Heng Huat |year=1999 |title=Ramanujan and the modular j-invariant |journal=] |volume=42 |issue=4 |pages=427–440 |doi=10.4153/CMB-1999-050-1 |mr=1727340 |s2cid=1816362 |zbl= |doi-access=free}}</ref> <math display="block">j(\tau) = q^{-1} + 744 + 196\,884 q + 21\,493\,760 q^2 + 864\,299\,970 q^3 + \cdots</math>
The ] holds as a ] ],

<math display=block>j(\tau) = q^{-1} + 744 + 196\,884 q + 21\,493\,760 q^2 + 864\,299\,970 q^3 + \cdots</math>

where <math>q = e^{2\pi i\tau}</math> and <math>\tau</math> the ] of an ].<ref>{{Cite journal |doi=10.4153/CMB-1999-050-1 |doi-access=free |first1=Bruce C. |last1=Berndt |author1-link=Bruce C. Berndt |first2=Heng Huat |last2=Chan |title=Ramanujan and the modular j-invariant |journal=] |volume=42 |issue=4 |year=1999 |pages=427–440 |mr=1727340 |s2cid= 1816362 |zbl=}}</ref> The function without the constant term, <math>J(\tau) = j(\tau) - 744</math>, is the ] of the ] whose symmetries constitute the ], <math>\mathrm {F_1}</math>.<ref name="GebMatt">{{Cite journal |last=Gaberdiel |first=Matthias R. |author-link=Matthias Gaberdiel |title=Constraints on extremal self-dual CFTs |journal=] |publisher=] |number=11, 087 |year=2007 |volume=2007 |pages=10–11 |bibcode=2007JHEP...11..087G |doi=10.1088/1126-6708/2007/11/087 |arxiv=0707.4073 |mr=2362062 |s2cid=16635058 |doi-access=free }}</ref>


== Almost integers == == Almost integers ==
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:"In particular, {{math|1=4124 = 3875 + 248 + 1 }} and {{math|1=34752 = 30380 + 3875 + 2 · 248 + 1 }}, where 248, 3875 and 30380 are all dimensions of irreducible representations of {{math|1='''E'''<sub>8</sub>(<math>\mathbb{C}</math>) }}."{{rp|6}}</ref>{{rp|p.7,9–11}} :"In particular, {{math|1=4124 = 3875 + 248 + 1 }} and {{math|1=34752 = 30380 + 3875 + 2 · 248 + 1 }}, where 248, 3875 and 30380 are all dimensions of irreducible representations of {{math|1='''E'''<sub>8</sub>(<math>\mathbb{C}</math>) }}."{{rp|6}}</ref>{{rp|p.7,9–11}}
:<math>\chi_{e_{8} \oplus e_{8} \oplus e_{8}} (q) = J(q) + 744 = j(q),</math> where the ] probabilistic ] for <math>\mathrm {F_{1}}</math> is <math>J(q)</math> of character <math>\chi_{F_{1}}.</math><ref name="GebMatt"/> :<math>\chi_{e_{8} \oplus e_{8} \oplus e_{8}} (q) = J(q) + 744 = j(q),</math> where the ] probabilistic ] for <math>\mathrm {F_{1}}</math> is <math>J(q)</math> of character <math>\chi_{F_{1}}.</math><ref name="GebMatt">{{Cite journal |last=Gaberdiel |first=Matthias R. |author-link=Matthias Gaberdiel |year=2007 |title=Constraints on extremal self-dual CFTs |journal=] |publisher=] |volume=2007 |pages=10–11 |arxiv=0707.4073 |bibcode=2007JHEP...11..087G |doi=10.1088/1126-6708/2007/11/087 |mr=2362062 |s2cid=16635058 |doi-access=free |number=11, 087}}</ref>


In the form of a ], the ] '''VOA''' is the first aside from <math>V_1</math> (as <math>\mathbb {C}_{24}</math>) with a ] <math>c</math> of <math>24</math>, out of a total seventy-one such modular invariant ] of ] VOAs of weight one.<ref>{{Cite journal |last1=Schellekens |first1=Adrian Norbert |title=Meromorphic ''c'' = 24 conformal field theories. |url=https://link.springer.com/article/10.1007/BF02099044 |journal=] |publisher=] |volume=153 |issue=1 |year=1993 |pages=159–185 |location=Berlin |doi=10.1007/BF02099044 |arxiv=hep-th/9205072 |bibcode=1993CMaPh.153..159S |mr=1213740 |zbl=0782.17014 |s2cid=250425623 }}</ref> Known as '']'', these algebras form ]s in <math>V_{\Lambda_{24}}</math> whose corresponding ]s are ] to the ] <math>V_{2}</math><sup>{{large|♮}}</sup> that contains <math>\mathrm {F_{1}}</math> as its automorphism;<ref>{{Cite journal |last1=Möller |first1=Sven |last2=Scheithauer |first2=Nils R. |title=Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra. |url=https://annals.math.princeton.edu/2023/197-1/p04 |journal=] |volume=197 |issue=1 |publisher=] & the ] |year=2023 |pages=261–285 |doi=10.4007/annals.2023.197.1.4 |bibcode=2019arXiv191004947M |mr=4513145 |zbl=1529.17040 |arxiv=1910.04947 |s2cid=204401905 }}</ref> of these, the second and third largest contain ] <math>E^3_{8,1}</math> and <math>D_{16,1}E_{8,1}</math> that are realized in <math>\operatorname {dim}744</math>. In the form of a ], the ] '''VOA''' is the first aside from <math>V_1</math> (as <math>\mathbb {C}_{24}</math>) with a ] <math>c</math> of <math>24</math>, out of a total seventy-one such modular invariant ] of ] VOAs of weight one.<ref>{{Cite journal |last1=Schellekens |first1=Adrian Norbert |title=Meromorphic ''c'' = 24 conformal field theories. |url=https://link.springer.com/article/10.1007/BF02099044 |journal=] |publisher=] |volume=153 |issue=1 |year=1993 |pages=159–185 |location=Berlin |doi=10.1007/BF02099044 |arxiv=hep-th/9205072 |bibcode=1993CMaPh.153..159S |mr=1213740 |zbl=0782.17014 |s2cid=250425623 }}</ref> Known as '']'', these algebras form ]s in <math>V_{\Lambda_{24}}</math> whose corresponding ]s are ] to the ] <math>V_{2}</math><sup>{{large|♮}}</sup> that contains <math>\mathrm {F_{1}}</math> as its automorphism;<ref>{{Cite journal |last1=Möller |first1=Sven |last2=Scheithauer |first2=Nils R. |title=Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra. |url=https://annals.math.princeton.edu/2023/197-1/p04 |journal=] |volume=197 |issue=1 |publisher=] & the ] |year=2023 |pages=261–285 |doi=10.4007/annals.2023.197.1.4 |bibcode=2019arXiv191004947M |mr=4513145 |zbl=1529.17040 |arxiv=1910.04947 |s2cid=204401905 }}</ref> of these, the second and third largest contain ] <math>E^3_{8,1}</math> and <math>D_{16,1}E_{8,1}</math> that are realized in <math>\operatorname {dim}744</math>.

Revision as of 19:56, 9 August 2024

Natural number
← 743 744 745 →
← 0 100 200 300 400 500 600 700 800 900 →
Cardinalseven hundred forty-four
Ordinal744th
(seven hundred forty-fourth)
Factorization2 × 3 × 31
Divisors1, 2, 3, 4, 6, 8, 12, 24, 31, 62, 93, 124, 186, 248, 372, 744
Greek numeralΨΜΔ´
Roman numeralDCCXLIV, dccxliv
Binary10111010002
Ternary10001203
Senary32406
Octal13508
Duodecimal52012
Hexadecimal2E816

744 (seven hundred 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 ! {\displaystyle k!} , as the sum of four consecutive primes p {\displaystyle p} , and as the product of sums of divisors σ ( n ) {\displaystyle \sigma (n)} of consecutive integers n {\displaystyle n} ; respectively:

744 = 4 ! + 6 ! 744 = 179 + 181 + 191 + 193 744 = σ ( 15 ) × σ ( 16 ) = 24 × 31 {\displaystyle {\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. It is also an abundant number, because the sum of its proper divisors is greater than itself.

The number partitions of the square of seven (49) into prime parts is 744.

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 φ ( n ) {\displaystyle \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 σ ( n ) {\displaystyle \sigma (n)} , where 744 sets the twenty-ninth record for σ ( n ) , {\displaystyle \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 {\displaystyle n} ) at seven hundred forty-four is equal to λ ( 744 ) = 30 = 2 × 3 × 5 {\displaystyle \lambda (744)=30=2\times 3\times 5} .

744 is the sixth number n {\displaystyle n} whose totient value has a sum-of-divisors equal to n {\displaystyle n} . In total, only seven numbers have sums of divisors equal to 744. Only one number has an aliquot sum that is 744, it is 456.

In graph theory

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

Properties specific to particular bases

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

Meanwhile, in septenary 744 is palindromic: 74410 = 21127.

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 } {\displaystyle \{1,2,3,...,11\}} with no consecutive integers.

j-invariant

The j–invariant can be written as a q–expansion Fourier series. j ( τ ) = q − 1 + 744 + 196 884 q + 21 493 760 q 2 + 864 299 970 q 3 + ⋯ {\displaystyle j(\tau )=q^{-1}+744+196\,884q+21\,493\,760q^{2}+864\,299\,970q^{3}+\cdots }

Almost integers

Ramanujan's constant is the transcendental almost integer

e π 163 ≈ 262 537 412 640 768 743.999 999 999 999 250 072 59 ≈ 640 320 3 + 744. {\displaystyle 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 π d {\displaystyle e^{\pi {\sqrt {d}}}} turn out to be nearly integers for special values of d {\displaystyle d} :

e π 19 ≈ 000 0 96 3 + 744 − 0.22 e π 43 ≈ 000 960 3 + 744 − 0.000 22 e π 67 ≈ 00 5 280 3 + 744 − 0.000 0013 e π 163 ≈ 640 320 3 + 744 − 0.000 000 000 000 75 {\displaystyle {\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 {\displaystyle 744} is theta series coefficient 25 {\displaystyle 25} of the four-dimensional cubic lattice D 4 ≅ ( Z 4 ) + {\displaystyle \mathbb {D} _{4}\cong \mathbb {(Z^{4})^{+}} } , or equivalently, the number of Hurwitz integer quaternions with norm 25. For the theta series of D 4 {\displaystyle \mathbb {D} _{4}} , that is realized in the 16-cell honeycomb, all 25 × 2 indexed coefficients (i.e. 25, 50, 100, 200, 400, ...) are 744.

E8 and the Leech lattice

The exceptional Lie algebra e 8 {\displaystyle {\mathfrak {e_{8}}}} has a graded dimension j ( q ) 1 / 3 {\displaystyle j(q)^{1/3}} whose character χ {\displaystyle \chi } lends to a direct sum equivalent to,

χ e 8 ⊕ e 8 ⊕ e 8 ( q ) = J ( q ) + 744 = j ( q ) , {\displaystyle \chi _{e_{8}\oplus e_{8}\oplus e_{8}}(q)=J(q)+744=j(q),} where the CFT probabilistic partition function for F 1 {\displaystyle \mathrm {F_{1}} } is J ( q ) {\displaystyle J(q)} of character χ F 1 . {\displaystyle \chi _{F_{1}}.}

In the form of a vertex operator algebra, the Leech lattice VOA is the first aside from V 1 {\displaystyle V_{1}} (as C 24 {\displaystyle \mathbb {C} _{24}} ) with a central charge c {\displaystyle c} of 24 {\displaystyle 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 Λ 24 {\displaystyle V_{\Lambda _{24}}} whose corresponding orbifold constructions are isomorphic to the moonshine module V 2 {\displaystyle V_{2}} that contains F 1 {\displaystyle \mathrm {F_{1}} } as its automorphism; of these, the second and third largest contain affine structures E 8 , 1 3 {\displaystyle E_{8,1}^{3}} and D 16 , 1 E 8 , 1 {\displaystyle D_{16,1}E_{8,1}} that are realized in dim ⁡ 744 {\displaystyle \operatorname {dim} 744} .

Other relationships

  • 744 is also the sum of consecutive pentagonal numbers: P 11 + P 12 + P 13 = 210 + 247 + 287. {\textstyle P_{11}+P_{12}+P_{13}=210+247+287.}
  • 744 is the second-smallest magic constant of a six by six magic square consisting of thirty-six consecutive prime numbers, between 41 and 223 inclusive.
  • 744 is the number of non-congruent polygonal regions in a regular 36–gon with all diagonals drawn.
  • There are 744 ways in-which fourteen squares of different sizes fit edge-to-edge inside a larger rectangle.

See also

References

  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.
  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.
  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 A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. 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.
  17. Sloane, N. J. A. (ed.). "Sequence A029954 (Palindromic in base 7.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-30.
  18. 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.
  19. 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.
  20. 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)
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. Sloane, N. J. A. (ed.). "Sequence A000118 (Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-12-23.
  28. 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.
  29. 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 · 248 + 1, where 248, 3875 and 30380 are all dimensions of irreducible representations of E8( C {\displaystyle \mathbb {C} } )."
  30. 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.
  31. 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.
  32. 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.
  33. 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.
  34. Sloane, N. J. A. (ed.). "Sequence A000326 (Pentagonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
  35. 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.
  36. 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.
  37. 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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