Tag Archives: reciprocal sum

2010/10/14 · 20:46

3, 7, 15, 23, 39, 47, 71, 87, 111, 127, …

10 Siehler’s numbers

two consecutive reciprocal partial sums are

$1/3+1/7+1/15+1/23+1/39+1/47+1/71+1/87+1/111+1/127+1/167+1/183+1/231+1/255+1/287+1/319+1/383+1/407+1/479+1/511+1/559+1/599+1/687+1/719+1/799+1/847+1/919+1/967+1/10794$

$\approx 0.722936$

and

$1/3+1/7+1/15+1/23+1/39+1/47+1/71+1/87+1/111+1/127+1/167+1/183+1/231+1/255+1/287+1/319+1/383+1/407+1/479+1/511+1/559+1/599+1/687+1/719+1/799+1/847+1/919+1/967+1/1079+$

$1/1111 \approx 0.723836$

…

(04.nov.2011)

by the way: $f(n)=\#{\mathbb{Z}_n}^*$ (cardinal of multiplicatively invertibles elements in the ring ${\mathbb{Z}_n}$ ) satisfy:

$f(1)=\#{\mathbb{Z}_1}^*=0$ , since ${\mathbb{Z}_1}=\{0\}$ and ${\mathbb{Z}_1}^*=\varnothing$
$f(2)=\#{\mathbb{Z}_2}^*=1$ , since ${\mathbb{Z}_2}=\{0,1\}$ and ${\mathbb{Z}_2}^*=\{1\}$
$f(3)=\#{\mathbb{Z}_3}^*=2$ , since ${\mathbb{Z}_3}=\{0,1,2\}$ and ${\mathbb{Z}_3}^*=\{1,2\}$
$f(4)=\#{\mathbb{Z}_4}^*=2$ , since ${\mathbb{Z}_4}=\{0,1,2,3\}$ and ${\mathbb{Z}_4}^*=\{1,3\}$
$f(5)=\#{\mathbb{Z}_4}^*=4$ , since ${\mathbb{Z}_5}=\{0,1,2,3,4\}$ and ${\mathbb{Z}_4}^*=\{1,2,3,4\}$
…

Hence

$s(1)=3+4f(1)=3$

$s(2)=s(1)+4f(2)=7$

$s(3)=s(2)+4f(3)=15$

$s(4)=s(3)+4f(4)=23$

$s(5)=s(4)+4f(5)=39$

…

$s(n+1)=s(n)+4f(n+1)$

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Tagged as A171503, coprimes, cucei, euler phi, eulerphi, number theory, OEIS, reciprocal sum, sequence, Siehler numbers, sum of reciprocals, totient funcion

2010/10/14 · 19:19

another reciprocal sum problem

ok we already know how to calculate how many rationals, $\mathbb{Q}$ , exists in terms of the height function

$h(a/b)=\max (|a|,|b|)$

$gcd(a,b)=1$

It turns out that the A171503 entry of the OEIS tells more…

http://192.20.225.10/~njas/sequences/A171503

meanwhile it is natural to ask for convergence of

$1/3 + 1/7 + 1/15 + 1/23 + 1/31 + 1/47 + 1/71 + 1/87 + 1/111 + 1/127+...$

and for (non-) existence of a generating function…

The crescent sequence could be called Siehler’s numbers

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Tagged as A171503, OEIS, reciprocal sum, reciprocal sums, reciprocal sums and series, Siehler numbers, sum of reciprocals

2010/09/26 · 21:34

reciprocal Catalan numbers

the subject that gives the grade of B.Sc. to my ex-pupil and friend, was originated making a conjecture that in less than of a month we will know how to demonstrate the formula in

http://www.research.att.com/~njas/sequences/A121839

but we were even able of finding a generating function for them

Congratulations David!

… this function is:

or $\frac{2\sqrt{4-x}(8+x)+12\sqrt{x}\arctan{\frac{\sqrt{x}}{\sqrt{4-x}}}}{\sqrt{(4-x)^5}}$

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Tagged as Catalan numbers, generating function, reciprocal sum, reciprocal sums and series, sum of reciprocals

2009/10/01 · 22:28

congreso nacional de la Sociedad Matemática Mexicana

Esta próxima semana del 12-16 de octubre tendremos Congreso Nacional (mexicano) de Matemáticas, el 42avo y en la ciudad de Zacatecas, además me es grato comunicarles que el CUCEI tiene presencia con al menos 3 ponencias:

Sum of reciprocal central binomial coefficients por David Íñiguez Báez
Geometría diferencial de superficies con la derivada covariante por Juan M. Márquez
Surface bundles over the circle por Juan M. Márquez

La primera y la tercera son reportes de investigación de correspondientes trabajos de tesis, éstas en las sesiones de combinatoria y matemáticas discretas y la otra en la sesiones de topología algebraica, respectivamente. David va a hablar sobre las relaciones que se dan acerca de las sumas de recíprocos de entre dos importantes secuencias crecientes de números en la combinatoria: números de Catalan y los coeficientes binomiales centrales. Juan hablará del estudio topológico de una variedad de tres dimensiones que surge dentro de los objetos conocidos como fibrados de Seifert (Seifert bundles) y que resulta tienen una característica especial: contiene un circle completo de singularidades en la correspondiente superficie de órbitas.

La segunda es una plática de divulgación acerca de la forma en que, acá en el CUCEI, estudiamos una parte de la geometría diferencial. Ésta, dentro de las sesiones de geometría y geometría algebraica.

El programa recientemente (1 de octubre) ha visto la luz y ya estabamos medio desesperados pues la publicación de esta información ya se había tardado mucho, en fin ya esta ahí y ahora habrá que planear a que cosas vamos a asistir en ese baquete intelectual.

¿Quieres saber qué más habrá allá? échale un ojo aquí

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Filed under Catalan numbers, differential geometry, mathematics, sum of reciprocals, what is math

Tagged as 3-manifolds, Congreso Nacional de Matemáticas, cucei, cucei math, reciprocal sum, Seifert bundle

2009/07/29 · 18:54

whole sum of the reciprocal Catalan numbers

In this little post I complete the details of the calculation

$2+\frac{4\sqrt{3}\pi}{27}=\sum_{n=0}^{\infty}\frac{n+1}{{2n\choose n}}=1+1+\frac{1}{2}+\frac{1}{5}+\frac{1}{14}+\frac{1}{42}+\cdots$

for the reciprocals $C_n=\frac{1}{n+1}{2n\choose n}$ , the famous so called Catalan numbers. It seems this is well known but it is scarcely quoted anywhere: Cf1, Cf2

We begin by recalling the relation ${C_n}^{-1}=(2n+1)(n+1)\int_0^1t^n(1-t)^n{\rm{d}}t$ , that is, in terms of the Beta function. So

$\sum_{n=0}^{\infty}{C_n}^{-1}=\sum_{n=0}^{\infty}(2n+1)(n+1)\int_0^1t^n(1-t)^n{\rm{d}}t$

$=\int_0^1\left[\sum_{n=0}^{\infty}(2n+1)(n+1)t^n(1-t)^n\right]{\rm{d}}t$

$=\int_0^1\frac{1+3t - 3t^2}{{(1-t+t^2)}^3}{\rm{d}}t$

$=2+\frac{4\sqrt{3}\pi}{27}$

This was hinted to me, thanks to, by the professor Qiaochu Yuan and I am not ashamed to confess that the calculations were executed in mathematica-v5. Why? well, the risk of introducing errors is a little-big.

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Tagged as A000108, A121839, Beta function, Catalan numbers, numbers, OEIS, reciprocal sum, sum of reciprocals