# Tagged Questions

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### Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ...
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### Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping $$f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$$ surjective? If not, what is its image? If yes, what can be said about ...
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### $- \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1}$ [closed]

Numerical evidence suggests: $$- \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1} \approx 0.764499780348444$$ Couldn't find cancellation via rearrangement. For the ...
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### Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence?

Question : Letting $k,n$ be positive integers, let's define a sequence $\{a_i\}\ (i=0,1,\cdots, kn)$ as $$(1+x+\cdots+x^k)^n=\sum_{i=0}^{kn}a_ix^i.$$ Then, is the 'special' difference-sequence ...
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### Convergence of $\sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$

Related to an open problem about another series. Set $$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$ where $\psi^{(n)}(k)$ is the polygamma function. Does $A$ converge? The related ...
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### recursive sum of products of bessel functions

I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$ recursively, i.e. once $S_0(x,y)$ is found, via the ...
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### Upper bounds for the sum of primes <= n

Let $s(n)$ denote the sum of primes less than or equal to n. Clearly, $s(n)$ is bounded from above by the sum of the first $n/2$ odd integers $+1$. $s(n)$ is also bounded by the sum of the first $n$ ...
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### Can an integer or rational sequence satisfy some bounded order recurrence $\mod \$ almost all primes but doesn't satisfy such in $\mathbb{Q}$?

Can an integer or rational sequence satisfy some bounded order recurrence $\mod \$ almost all primes but doesn't satisfy such in $\mathbb{Q}$? The recurrences $\mod p$ can be different, possibly ...
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### Computability of OEIS A034891 …partitions of n into prime parts (1 included)

On the seqfan mailing list RGWv gave short algorithm for computing A000041 number of partitions of n the partition numbers: f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == ...
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### Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$: For each positive ...
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### mertens-function in the light of divergent summation - what summation method were best adapted

Just reading about the Mertens-function in the other thread Mertens function I remember an earlier attempt to apply divergent summation to the series which is constructed of the Moebius-function at ...