All Questions
Tagged with sequences-and-series number-theory
1,553
questions
2
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65
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Conjecture on Infinitely Many Consecutive Pairs of Early Primes
An early prime is one which is less than the arithmetic mean of the prime before and the prime after.
Conjecture: There are infinitely many consecutive pairs of early primes
MY attempt
Well, the fact ...
- 59
4
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1
answer
135
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2-adic valuations of $k\cdot 3^n-1$
I was just playing around with numbers of the form $3^n-1$, and noticed that their 2-adic valuations have a nice, understandable pattern:
$\left(v_2(3^n-1)\right)_{n\ge 1} = (1,3,1,4,1,3,1,5,1,3,1,4,1,...
- 31.5k
1
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61
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Lower bound of Hypergeometric function ${}_2F_1(d,1,d+1, z)$ [closed]
I am looking for a (non-trivial) lower bound of the Gauss hypergeometric function ${}_2F_1(d,1,d+1, z)$ where $d\in\mathbb{N}$ and $0\leq z <1$. Ideally, the bound would be valid $\forall d\in\...
- 11
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1
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Termwise Differentiation of Convergent Series Involving the Riemann Zeta Function
Let $\zeta(s)$ be the Riemann zeta function. We know that
$$\zeta(s) = \prod_p(1 - p^{-s})^{-1}, \ \operatorname{Re} s > 1$$
in the sense that the righthand side converges absolutely to $\zeta(s)$ ...
- 4,429
1
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30
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Existence of $n$ where $S_b(n^k) \equiv r \pmod{M}$ where $S_b$ denotes sum of digits in base $b$
Let $b, k, M \in \mathbb{N} \setminus \{1\}$, $r \in \{0, 1, \dots M-1\}$ and $S_b: \mathbb{N}_0 \rightarrow \mathbb{N}_0 $ denote the method which outputs the sum of digits of its input in base $b$.
...
- 1,181
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Is the non-uniform membership problem for sets obtained from 1 by applying affine integer functions P/NP-complete/other?
The question concerns algorithmic complexity of the membership problem for sets obtained from $1$ by applying a fixed number of affine functions. One such example is Klarner-Rado sequence (A002977 in ...
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3
votes
1
answer
84
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For which integers $m$ does an infinite string of characters $S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$ exist
Question:
For which integers $m$ does an infinite string of characters
$$S = c_{1} \cdot c_{2} \cdot c_{3} \cdot c_{4} \cdot c_{5} \ldots$$
exist such that for all $n \in \mathbb{Z}_{>0}$ there are ...
2
votes
1
answer
104
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Define the sum of transfinite ordinal sequences according to finite ordinal sum.
I know it is possibile to define finite sum of ordinals: if $(\alpha_i)_{i\in n}$ is a sequence of lenght $n$, with $n$ in $\omega$, then the symbolism $\sum_{i\in n}\alpha_i$ has a (formal) meaning; ...
- 4,906
2
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0
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139
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Divisors sum and Bessel Function related sums
Discovered the following relation:
$$\sum _{k=1}^{\infty } \sigma (k) \left(K_2\left(4 \pi \sqrt{k+y} \sqrt{y}\right)-K_0\left(4 \pi \sqrt{k+y} \sqrt{y}\right)\right)=\frac{\pi K_1(4 \pi y)-3 K_0(...
- 1,818
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75
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$1^2+2^2+3^2+\cdots+n^2=y^2$ ($n$ and $y$ are natural numbers) [duplicate]
I tried solving this so first $1^2+2^2+\cdots+n^2=n(n+1)(2n+1)/6$ so problem is $n(n+1)(2n+1)=6y^2$ where $n$ and $y$ are natural. I tried doing this $n=6k,6k+1....6k+5$, so say for $n=6k$
you get $k(...
3
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0
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Formula including divisors sum ($\sigma$), Euler Gamma ($\gamma$), $\pi$ and $\ln \pi$ [closed]
An interesting formula arose during the investigation of divisors sum efficient calculation. Actually the below series converges very slowly, as every series containing $\gamma$ :)
$$\sum _{k=1}^{\...
- 1,818
6
votes
1
answer
95
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How do I check if this number is transcendental?
Two days ago, I tried to create an infinite series that might be able to generate a transcendental number, and when I checked the proper definition, it was mentioned that, it is a number that cannot ...
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2
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1
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77
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Let $x_0 = 3;\ x_{n+1}=3x_n\ $ if $\ \frac{x_n}{2}<1;\ x_{n+1}=\frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\ $ Is $\ \liminf_{n\to\infty} x_n=1?$
This is a natural follow-up question of this previous question of mine.
Let $x_0 = 3.$ Let $\ x_{n+1} = 3x_n\ $ if $\ \frac{x_n}{2}<1;\quad x_{n+1} = \frac{x_n}{2}\ $ if $\ \frac{x_n}{2}>1.\quad ...
- 20.7k
1
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31
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Looking for a solution to a 2-dimensional recurrence relation with a floor function in GF(2)
I have a recurrence relation that i'd like to solve. It is defined as follows:
$$ a_{i,j} = (a_{i-1,j} + \lfloor 3 \sum_{k=0}^{j-1} \frac{a_{i-1,k}}{2^{j-k}} \rfloor \space) \bmod 2 $$
where $ a_{i,...
2
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Except for $1$, are there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$?
Other than $1$, is there any perfect powers in the sequence $1,122,122333,\cdots,122333\cdots\underbrace{nnn\cdots n}_{n \text{ times}}$?
This is sequence A098129.
We certainly know that this is not a ...
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