As stated in A123323: Number of triples a,b,c with a<=b<=c<a+b, gcd(a,b,c)=1 and c=n. …… A123323(n)=sumdiv(n, d, floor((d+1)^2/4)*moebius(n/d)) How to get this formula and how Moebius transformation works? Thanks.

Let $n_k$ be integers. Is there a general formula for the Taylor expansion of $\log(1+\sum_{k=1}^N n_kx^k)$ at $x=0$? This boils down to find an expression for the $m$th derivative of $\log(1+\sum_{k=1}^N n_kx^k)$ evaluated $x=0$: $$ \frac{d^m}{dx^m}\log(1+\sum_{k=1}^N n_kx^k)\Biggr|_{x=0}? $$ Expanding a (work-in-progress) example like $-\log(n_3x^3+n_2x^2+n_1x+1)$ gives: $$ \begin{array}{cl} x& (- n_1)\\ + \frac{x^2}2& (+ n_1^2 – […]

This question is about the integer sequence A123168. The title of the sequence reads: Continued fraction for – $$\sqrt{2}\Bigg({e^\sqrt{2}-1 \above 1.5pt e^\sqrt{2}+1 }\Bigg)$$ A comment claims that ‘this continued fraction shows that $e^\sqrt{2}$ is irrational”. I thought Lindemann & Weierstrass theorem established the irrationality of $e^\sqrt{2}$ so what is the motivation behind this particular continued […]

This question is somewhat related to one of my previous questions: Fibonorial of a fractional or complex argument. Recall the definition of harmonic numbers: $$H_n=\sum_{k=1}^n\frac1k=1+\frac12+\,…\,+\frac1n\tag1$$ Obviously, harmonic numbers satisfy the following functional equation: $$H_n-H_{n-1}=\frac1n\tag2$$ The definition $(1)$ is valid only for $n\in\mathbb N$, but it can be generalized to all positive indices. There are several […]

Divide a square into non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? This is known as the Mondrian Art Problem. For example, here’s a division for a square of size 138. The largest area is 1200, the smallest 1178, with a […]

Consider the following iterative process. We start with a 2-element set $S_0=\{0,1\}$. At $n^{\text{th}}$ step $(n\ge1)$ we take all non-empty subsets of $S_{n-1}$, then for each subset compute the arithmetic mean of its elements, and collect the results to a new set $S_n$. Let $a_n$ be the size of $S_n$. Note that, because some subsets […]

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