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$\eth(n)$ is a little algorithm I made, which may appear to be quite complex, so I will start with an example middle of the post. Questions are at the end of the post.

Definition

Let $W$ be the function , defined as $W(a,b)=r$

Let's given $a,b\in \mathbb{N}$ and $a>1$

Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \lceil \log{b}/\log{a} \rceil - 1$.

Arrange as: $$a^{m+1} - b$$ $$ = r_{l} a^l + r_{l-1} a^{l-1}+... + r_1 a^1 + r_0 a^0 $$ $$=(r_{l} r_{l-1} ... r_{1} r_{0})_{a}$$

Where $r=\sum_{i=0}^{l}r_{i}$

Note $W(a,0)=$ not define

Now here $n \in \mathbb{N}$

$S$ is a function defined as

$$S(a,n)=\sum_{i=1}^{a}i^{n}$$

Let's $p$ is prime and $p+1=z$

$\eth$ is a function defined as

$$\eth (n) = \sum_{W(z,W(z,S(z,2n)))\ne z}1$$

I strongly observed

If $ z>2n+2$ Then $W(z,W(z,S(z,2n)))=z$

Hence I conclude

$$\eth(n)\leq \pi (2n+1)$$

$$\eth(n) \approx \pi (n)$$

$$|\eth(n) - \pi (n)|\leq 2$$

Observation table

$$\begin{array}{c | c | c |c | } n & \eth(n) & \pi(n) \\ \hline 1 & 2 & 0 \\ \hline 2 & 3 & 1 \\ \hline 3 & 3 & 2 \\ \hline 5 &4& 3 \\ \hline 9 &4& 4 \\ \hline 10 &5& 4 \\ \hline 50 &15& 15 \\ \hline 100 &26& 25 \\ \hline 200 &44& 46 \\ \hline \end{array}$$

Example

we want to find $W(6,W(6,S(6,2)))$

First calculate $S(6,2)=1^{2}+2^{2}+...+6^{2}=91$

$\implies W(6,W(6,91))$

Here for calculate $W(6,91)$

$ 6^{3}-91 = 125 = (325)_{6}$

$\implies r = \sum r_{i} = 3+2+5 =10$

$hence W(6,91) = 10$

Again to calculate $W(6,W(6,91))=W(6,10)$

$6^{2}-10 =26 = (42)_{6}$

$\implies r=\sum r_{i} = 4+2 =6$

Hence $W(6,W(6,S(6,2)))=6$

Table For $W(t,W(t,S(t,2)))$ which helps to calculate $\eth(1)$.

$$\begin{array}{c | c | c |c | } t & W(t,S(t,2)) & W(t,W(t,S(t,2))) \\ \hline 2 & 2 & 0 \\ \hline 3^{*} & 3 & 0 \\ \hline 4^{*} & 4 & 0 \\ \hline 5 & 6 & 7 \\ \hline 6^{*} & 10 & 6 \\ \hline 7 &5 & 2 \\ \hline 8^{*} &14& 8 \\ \hline 9 &12& 13 \\ \hline 10 &12& 16 \\ \hline 11 & 15 & 16 \\ \hline 12^{*} & 22 & 12 \\ \hline 13 & 10 & 3 \\ \hline 14^{*} & 26 & 14 \\ \hline 15 & 21 & 22 \\ \hline 16 &20 & 26 \\ \hline 17 &24& 25 \\ \hline 18^{*} &34& 18 \\ \hline 19 &15& 4 \\ \hline 20^{*} &38& 20 \\ \hline 21 &30& 31 \\ \hline \vdots &\vdots & \vdots \\ \hline \end{array}$$

$t^{*} = z $

From table $W(t,W(t,S(t,2)))$ we can calculate $\eth(1)$ by counting $z$ such that $W(z,W(z,S(z,2)))\ne z$. we can observe it's only happens when $z=3$ and $4$ hence $\eth(1)= 2 $

●We can prove easily

$W(a+1,ax+i)=ay+j$ iff $i+j\equiv 1($ mod $a)$ for $0\leq i,j < a$

$\chi$ is function defined as

$$\chi(n)=\sum_{p \nmid S(p,2n)}1=\sum_{p-1|2n}1$$

And $ n \in \mathbb{N}$

Proof for $\chi(n)$

$\implies \eth (n)\geq \chi(n)$

Question

  • What is formula for $\eth(n)$?

  • Can we prove above observation ?

Note

● if $W(a,(a-1)q)=a$

Then $q\in$ {$a^{n}+(-1)^ni | i=${$1,2,..,(a+n-2)$}$\forall n \in \mathbb{W}$}

*I think ,$\eth$ function very much impact on prime number theorem, thank you.

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    $\begingroup$ $\Lambda$ and $\Omega$ both have standard meanings in Analytic Number Theory, so they are not good choices for names of your functions. $\endgroup$ – Gerry Myerson Jun 26 at 2:54
  • $\begingroup$ @Gerry_Myerson Yes you are right now it's changed $\endgroup$ – Pruthviraj Jul 12 at 14:12
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    $\begingroup$ Take a look at the other questions to see how you need to improve yours (simplify, shorter, motivated) $\endgroup$ – reuns Jul 12 at 21:10

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