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Relevant bibliographies by topics / Hardy-Ramanujan Number

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Academic literature on the topic 'Hardy-Ramanujan Number'

Author: Grafiati

Published: 5 June 2025

Last updated: 15 July 2025

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Journal articles on the topic "Hardy-Ramanujan Number"

1

Darbari, Mita. "A Connection between Hardy-Ramanujan Number and Special Pythagorean Triangles." Bulletin of Society for Mathematical Services and Standards 10 (June 2014): 45–47. http://dx.doi.org/10.18052/www.scipress.com/bsmass.10.45.

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Almkvist, Gert, and George E. Andrews. "A Hardy-Ramanujan formula for restricted partitions." Journal of Number Theory 38, no. 2 (1991): 135–44. http://dx.doi.org/10.1016/0022-314x(91)90079-q.

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Hossain, Md Fazlee, Nil Ratan Bhattacharjee, and Sabuj Das. "Ramanujan’s Famous Partition Congruences." Asian Journal of Applied Science and Engineering 5, no. 1 (2016): 145–58. http://dx.doi.org/10.18034/ajase.v5i1.74.

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In 1742, Firstly Leonhard Euler invented the generating function for , where is the number of partitions of n [ is defined to be 1]. Srinivasa Ramanujan was born on 22 December 1887. In 1916, S. Ramanujan invented the generating function for (2nd time). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of . MacMahon established a table of for the first 200 values of n, and Ramanujan observed that the table indicated certain simple congruences properties of . In 1916, S. Ramanujan quoted his famous partition congrucnecs. In particular, the numbers of the partitions of numbers 5m+ 4, 7m+5, and 11m +6 are divisible by 5, 7, and 11 respectively. Now this paper shows how to prove the Ramanujan’s famous partitions congruences modulo 5, 7, and 11 respectively.

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Gupta, Chirag. "Eulers Totient Function (Number Theoretic Functions), Right Angle Triangle and Their Applications." International Journal for Research in Applied Science and Engineering Technology 10, no. 1 (2022): 178–89. http://dx.doi.org/10.22214/ijraset.2022.39772.

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Abstract: In this paper, we introduce some new theorem and results(section Ⅱ,Ⅲ and Ⅵ) on Euler’ s Totient Function , Right angle triangle and their applications Keywords: Euler’ s totient function, right angle triangle, Golden ratio, Hardy – Ramanujan number, student’s pencil compass and etc.

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Charsooghi, Mohammad Avalin, Yousof Azizi, Mehdi Hassani, and Laleh Mollazadeh-Beidokhti. "On a Result of Hardy and Ramanujan." Sarajevo Journal of Mathematics 4, no. 2 (2024): 147–53. http://dx.doi.org/10.5644/sjm.04.2.01.

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In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}\Omega(k)$, where $\Omega(k)$ is the total number of prime factors of $k$. 2000 Mathematics Subject Classification. 05A10, 11A41, 26D15, 26D20

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6

Sills, Andrew V. "A Rademacher Type Formula for Partitions and Overpartitions." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–21. http://dx.doi.org/10.1155/2010/630458.

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A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of and the Zuckerman formula for the Fourier coefficients of is presented.

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Pollack, Paul. "A generalization of the Hardy–Ramanujan inequality and applications." Journal of Number Theory 210 (May 2020): 171–82. http://dx.doi.org/10.1016/j.jnt.2019.10.004.

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8

Murty, M. Ram, and S. Saradha. "On the Sieve of Eratosthenes." Canadian Journal of Mathematics 39, no. 5 (1987): 1107–22. http://dx.doi.org/10.4153/cjm-1987-056-8.

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Let v(n) denote the number of distinct prime factors of a natural number n. A classical theorem of Hardy and Ramanujan states that the normal order of v(n) is log log n. That is, given any , the number of natural numbers not exceeding x which fail to satisfy the inequality1is o(x) as x → ∞. A very simple proof of this was subsequently given by Turán. He showed that2

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9

Bayless, Jonathan, and Paul Kinlaw. "Consecutive coincidences of Euler’s function." International Journal of Number Theory 12, no. 04 (2016): 1011–26. http://dx.doi.org/10.1142/s1793042116500639.

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We prove a version of the Hardy–Ramanujan inequality and a bound on the count of smooth numbers up to some number [Formula: see text], both with explicit constants. We use these as tools to prove a few interesting results on the values [Formula: see text] satisfying [Formula: see text] and provide an explicit bound on the sum of the reciprocals of such [Formula: see text].

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Calude, Cristian S., Elena Calude, and Michael Dinneen. "What is the Value of Taxicab(6)?" JUCS - Journal of Universal Computer Science 9, no. (10) (2003): 1196–203. https://doi.org/10.3217/jucs-009-10-1196.

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For almost 350 years it was known that 1729 is the smallest integer which can be expressed as the sum of two positive cubes in two different ways. Motivated by a famous story involving Hardy and Ramanujan, a class of numbers called Taxicab Numbers has been defined: Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth powers in n different ways. So, Taxicab(3, 2, 2) = 1729, Taxicab(4, 2, 2) = 635318657. Computing Taxicab Numbers is challenging and interesting, both from mathematical and programming points of view. The exact value of Taxicab(6) = Taxicab(3, 2, 6) is not known, however, recent results announced by Rathbun [R2002] show that Taxicab(6) is in the interval [10 18 , 24153319581254312065344]. In this note we show that with probability greater than 99%, Taxicab(6) = 24153319581254312065344.

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Book chapters on the topic "Hardy-Ramanujan Number"

1

Pinsky, Ross G. "The Hardy–Ramanujan Theorem on the Number of Distinct Prime Divisors." In Universitext. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-07965-3_8.

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Effinger, Gove W., and David R. Hayes. "The polynomial Waring and Goldbach problems." In Additive Number Theory of Polynomials Over a Finite Field. Oxford University PressOxford, 1991. http://dx.doi.org/10.1093/oso/9780198535836.003.0001.

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Abstract From 1930 to 1932, Leonard Carlitz held a National Research Fellowship. He spent a substantial portion of his fellowship at Cambridge University, where he met Raymond Paley, a brilliant young English mathematician of about the same age as himself. In the decade just gone by, Hardy and Littlewood had captured the interest of the mathematical world with a remarkable series of papers on two classic problems in additive number theory—the Waring problem and the Goldbach problem. The new technique that they relentlessly focused on these problems was the circle method, first introduced by Hardy and Ramanujan in 1918 to derive an asymptotic expansion for the partition function.

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Wilson, Robin. "9. Partitions." In Combinatorics: A Very Short Introduction. Oxford University Press, 2016. http://dx.doi.org/10.1093/actrade/9780198723493.003.0009.

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How many ways can a number be split into two, three, or more pieces? ‘Partitions’ considers this interesting problem and the way in which Leonard Euler started to investigate them around 1740. Euler considered the generating function of the sequence of partition numbers and devised his pentagonal number formula. His publication Introduction to the Analysis of Infinities in 1748 outlined the difference between distinct and odd partitions. Many mathematicians worked on the partition problem, but it was not resolved until G. H. Hardy and his collaborator Srinivasa Ramanujan in 1918 published an exact formula for partition numbers using a new method in the theory of numbers called the ‘circle method’.

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Pickover, Clifford A. "A Ranking of the 5 Strangest Mathematicians Who Ever Lived." In Wonders of Numbers. Oxford University PressNew York, NY, 2001. http://dx.doi.org/10.1093/oso/9780195133424.003.0031.

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Abstract Srinivasa Ramanujan (1887-1920) Ramanujan, who started life as a clerk in the accounting department of the Madras post office, became India’s greatest mathematical genius and one of the greatest 20th-century mathematicians. Ramanujan made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. He came from a poor family, and his mother took in boarders, which created a crowded home. Ramanujan was very shy and found it hard to speak. He excelled in math but usually failed all his other courses. When he was 13, he borrowed a high school student’s math book and mastered it in a week. Because he was deprived of manuals that could teach him about rigorous proofs, Ramanujan developed rather strange methods to establish mathematical truths. Mathematician G. H. Hardy remarked:

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