On the Sums of Infinite and Finite Series of Reciprocals of Prime Factors and Divisors of the Number 2025

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F60162694%3AG43__%2F26%3A00566643" target="_blank" >RIV/60162694:G43__/26:00566643 - isvavai.cz</a>

Výsledek na webu

<a href="https://mitav.unob.cz/data/235_FVT_ISBN_642-8_e-kniha.pdf" target="_blank" >https://mitav.unob.cz/data/235_FVT_ISBN_642-8_e-kniha.pdf</a>

DOI - Digital Object Identifier

—

Jazyk výsledku

angličtina

Název v původním jazyce

On the Sums of Infinite and Finite Series of Reciprocals of Prime Factors and Divisors of the Number 2025

Popis výsledku v původním jazyce

This paper is inspired by one task on the sum of a special infinite series on the Berkeley Math Circle. The task was to determine the sum of all the unit fractions that have denominators with only factors consisting of all prime divisors of the number 2002, that is the primes 2, 7, 11 and 13. We updated this task to determine the sum of reciprocals of all products generated by all prime divisors of the number 2025. This series can be consider as the reduced harmonic series generated by two primes 3 and 5. The sum of this series is calculated analytically and finally also numerically by using the programming language of the computer algebra system Maple. By these both attempts we obtained the same result. The paper also includes a relatively easy determination of the sum of the reciprocals of all positive divisors of the number 2025, i.e. the sum of the inverted values of numbers 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675 and 2025. Finally, generalizations of both results are presented.

Název v anglickém jazyce

On the Sums of Infinite and Finite Series of Reciprocals of Prime Factors and Divisors of the Number 2025

Popis výsledku anglicky

This paper is inspired by one task on the sum of a special infinite series on the Berkeley Math Circle. The task was to determine the sum of all the unit fractions that have denominators with only factors consisting of all prime divisors of the number 2002, that is the primes 2, 7, 11 and 13. We updated this task to determine the sum of reciprocals of all products generated by all prime divisors of the number 2025. This series can be consider as the reduced harmonic series generated by two primes 3 and 5. The sum of this series is calculated analytically and finally also numerically by using the programming language of the computer algebra system Maple. By these both attempts we obtained the same result. The paper also includes a relatively easy determination of the sum of the reciprocals of all positive divisors of the number 2025, i.e. the sum of the inverted values of numbers 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675 and 2025. Finally, generalizations of both results are presented.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Mathematics, Information Technologies and Applied Sciences 2025 (post-conference proceedings of extended versions of selected papers)

ISBN

978-80-7582-642-8

ISSN

—

e-ISSN

—

Počet stran výsledku

10

Strana od-do

64-73

Název nakladatele

Univerzita obrany, Brno

Místo vydání

Brno

Místo konání akce

Brno

Datum konání akce

19. 6. 2025

Typ akce podle státní příslušnosti

EUR - Evropská akce

Kód UT WoS článku

—