Marie Juzmiyanti, 1017031034 (2015) KARAKTERISTIK BILANGAN TAU. FMIPA, UNIVERSITAS LAMPUNG.
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Abstract
Kennedy dan Cooper mendefinisikan bilangan bulat positif menjadi bilangan Tau jika , τ adalah fungsi banyaknya pembagi dari n. Beberapa bilangan Tau pertama antara lain : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, . . . ; yang merupakan barisan Sloane. Selain itu, Kennedy dan Cooper menunjukkan bahwa bilangan Tau mempunyai kepadatan nol. Konsep bilangan Tau ditemukan kembali oleh Colton, yang menyebutkan bahwa bilangan tersebut dapat difaktorkan kembali. Colton menduga bahwa bilangan Tau kurang dari atau sama dengan setengah dari banyaknya bilangan prima kurang dari atau sama dengan n. Selanjutnya Colton menduga bahwa untuk bilangan n yang cukup besar juga masih berlaku dengan membuktikan perumumannya. Juga dihitung batas bawah dari contoh 7.42 · 1013. Colton juga menduga bahwa tidak ada tiga bilangan Tau berurutan. Hasil lainnya adalah sifat bilangan Tau dibandingkan dengan bilangan prima. Juga dibahas beberapa perumuman bilangan Tau. Kata Kunci : Bilangan Tau , Bilangan Prima, Bilangan Bulat Positif CHARACTERISTIC OF TAU NUMBER Kennedy and Cooper defined a positive integer to be a tau number if , where τ is the number of divisors function. The first few Tau numbers are : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, . . . ; it is Sloane’s sequence. Among other things, Kennedy and Cooper showed the Tau numbers have density zero. The concept of Tau number was rediscovered by Colton, who called these numbers refactorable. This paper is primarily concerned with two conjectures made by Colton. Colton conjectured that the number of Tau numbers less than or equal to a given n was at least half the number of primes less than or equal to n. In this paper I show that Colton’s conjecture is true for all sufficiently large n by proving a generalized version of the conjecture. I calculate an upper bound for counter examples of . Colton also conjectured that there are no three consecutive Tau numbers. Other results are also given, including the properties of the Tau numbers as compared to the primes. Various generalizations of the Tau numbers are also discussed. Keywords : Tau Number, Prime Number, Positive Integer
| Item Type: | Other |
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| Subjects: | Q Science (General) > QA Mathematics |
| Divisions: | Fakultas MIPA > Prodi Matematika |
| Depositing User: | 7571580 . Digilib |
| Date Deposited: | 28 Dec 2015 06:51 |
| Last Modified: | 28 Dec 2015 06:51 |
| URI: | http://digilib.unila.ac.id/id/eprint/16272 |
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