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, 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.

In mathematics

744 is a semiperfect number. ^[1] It is also an abundant number. ^{[2] [3]}

The j-invariant, an important function in the study of modular forms and Monstrous moonshine, can be written as a Fourier series in which the constant term is 744: ^[4] $${\displaystyle j(\tau )=q^{-1}+744+196\,884q+21\,493\,760q^{2}+864\,299\,970q^{3}+\cdots ,}$$where ${\displaystyle q=e^{2\pi i\tau }}$. One consequence of this is that 744 appears in expressions for Ramanujan's constant and other almost integers.

References

1. Sloane, N. J. A. (ed.). "SequenceA005835(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.
2. Sloane, N. J. A. (ed.). "SequenceA005101(Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-04-03.
3. Sloane, N. J. A. (ed.). "SequenceA033880(Abundance of n, or (sum of divisors of n) - 2n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-12-29.
4. 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.